Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find a polynomial only from its roots Given $\alpha,\,\beta,\,\gamma$ three roots of $g(x)\in\mathbb Q[x]$, a monic polynomial of degree $3$. We know that $\alpha+\beta+\gamma=0$, $\alpha^2+\beta^2+\gamma^2=2009$ and $\alpha\,\beta\,\gamma=456$. Is it possible to find the polynomial $g(x)$ only from these?
I've been wo... | The polynomial is $(x-a)(x-b)(x-c)$ with the roots being $a,b,c$. By saying "three roots" you imply all these are different. Note that when multiplied out and coefficients are collected you have three symmetric functions in the roots. For example the constant term is $-abc$, while the degree 2 coefficient is $-(a+b+c)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple linear recursion $x_n=\frac{x_{n-1}}{a}+\frac{b}{a}$ with $a>1, b>0$ and $x_0>0$
I tried to solve it using the generating function but it does not work because of $\frac{b}{a}$, so may you have an idea.
| Hint: Let $x_n=y_n+c$, where we will choose $c$ later. Then
$$y_n+c=\frac{y_{n-1}+c}{a}+\frac{b}{a}.$$
Now can you choose $c$ so that the recurrence for the $y$'s has no pesky constant term?
Remark: There is a fancier version of the above trick. Our recurrence (if $b\ne 0$) is not homogeneous. To solve it, we find the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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List of interesting integrals for early calculus students I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely challenging like Putnam problems or anything). For example, the... | I remember having fun with integrating some step functions, for example:
$$\int_{0}^{2} \lfloor x \rfloor - 2 \left\lfloor \frac{x}{2} \right\rfloor \,\mathrm{d}x.$$
My professor for calculus III liked to make us compute piecewise functions, so it would force us to use the Riemann sum definition of the integral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/233162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "79",
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"answer_id": 2
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Finding a certain integral basis for a quadratic extension This is a problem in the first chapter of Dino Lorenzini's book on arithmetic geometry. Let $A$ be a PID with field of fractions $K$ and $L/K$ a quadratic extension (no separability assumption). Let $B$ be the integral closure of $A$ in $L$. Now assuming that $... | Consider the quotient $B/A$. What can you say about it?
| {
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"url": "https://math.stackexchange.com/questions/233219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Convergence properties of a moment generating function for a random variable without a finite upper bound. I'm stuck on a homework problem which requires me that I prove the following:
Say $X$ is a random variable without a finite upper bound (that is, $F_X(x) < 1$ for all $x \in \mathbb{R}$). Let $M_X(s)$ denote the m... | Consider the limit when $s\to+\infty$ of the inequality
$$
s^{-1}\log M_X(s)\geqslant x+s^{-1}\log(1-F_X(x)).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/233312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inscribed and Escribed Squares Assume a circle of diameter $d$. Inscribe a square $A$ centred in the circle with its diagonal equal to the diameter of the circle. Now escribe a square $B$ with the sides equal to the diameter of the circle. Show how to obtain the ratio of the area of square $A$ to the area of square $B$... | This can be done by a computation. The outer square $B$ has area $d^2$. Let the side of the inner square $A$ be $s$. Then by the Pythagorean theorem, $s^2+s^2=d^2$. But $s^2$ is the area if the inner square, and we are finished.
But there is a neater way! Rotate the inner square $A$ about the centre of the circle, unti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233399",
"timestamp": "2023-03-29T00:00:00",
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Extra $100 after borrowing and shopping I took \$1000 from my friend James and \$500 from Bond.
While walking to the shops I lost \$1000 so now I only have \$500.
I did some shopping, spending \$300 so now I have \$200 left.
I gave \$100 back to James and \$100 back to Bond.
Now my liabilities are \$900 for James and \... | That $\$1300$ already includes the $\$300$, along with the $\$1000$ you lost--that was your net loss of money for the day--you don't need to add it again.
The $\$900$ and the $\$400$ you still owe is just another way of reaching the same number.
| {
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"url": "https://math.stackexchange.com/questions/233476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Trace of a matrix to the $n$ Why is it that if $A(t), B(t)$ are two $n\times n$ complex matrices and $${d\over dt}A=AB-BA$$ then the trace of the matrix $A^n$ where $n\in \mathbb Z$ is a constant for all $t$?
| Note that Trace(FE)=Trace(EF) in general.
$n>0$ : Trace$(A^n)' = n [$Trace$ (A'(t) A^{n-1})] = n[ $Trace$ ((AB - BA)A^{n-1})] = 0$
$ n=0$ : $A^0 = I$ So we are done
$n <0$ : Check $(A^{-1})' = A^{-1} B - BA^{-1}$ So this case is reduced to the first case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/233577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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linear operator on a vector space V such that $T^2 -T +I=0$ let T be a linear operator on a vector space V such that $T^2 -T +I=0$.Then
*
*T is oneone but not onto.
*T is onto but not one one.
*T is invertible.
*no such T exists.
could any one give me just hint?
| $$
T^2-T+I=0 \iff T(I-T)=I=(I-T)T,
$$
i.e. $T$ is invertible and $T^{-1}=I-T$. In particular $T$ is injective and surjective.
| {
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"url": "https://math.stackexchange.com/questions/233666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the transcendence of the number $e$ How to prove that the number $e=2.718281...$ is a transcendental number? The truth is I have no idea how to do it.
If I can recommend a book or reference on this topic thank you.
There are many tests on the transcendence of $ e $?
I'd read several shows on the transcenden... | Your might be interested in the Lindemann-Weierstrass-theorem, which is useful for proving the transcendence of numbers, e.g., $\pi$ and $e$. If you read further, you'll see that the transcendence of both $\pi$ and $e$ are direct "corollaries" of the Lindemann-Weierstrass theorem.
Indeed, $e^x$ is transcendent if $x$ ... | {
"language": "en",
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Find the kernel of a linear transformation of $P_2$ to $P_1$ For some reason, this particular problem is throwing me off:
Find the kernel of the linear transformation:
$T: P_2 \rightarrow P_1$ $T(a_0+a_1x+a_2x^2)=a_1+2a_2x$
Since the kernel is the set of all vectors in $V$ that satisfy $T(\vec{v})=\vec{0}$, it's obvio... | Your argument is totally correct. Your book means that $\ker(T)=\{a_0+0\cdot x+0\cdot x^2|a_0\in \mathbb{R}\}$, i.e. $a_1=0$ and $a_2=0$, which is the same as you proved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/233781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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An inequality for $W^{k,p}$ norms Let $u \in W_0^{2,p}(\Omega)$, for $\Omega$ a bounded subset of $\mathbb R^n$. I am trying to obtain the bound
$$\|Du\|_p \leq \epsilon \|D^2 u\|_p + C_\epsilon \|u\|_p$$
for any $\epsilon > 0$ (here $C_\epsilon$ is a constant that depends on $\epsilon$, and $\|.\|_p$ is the $L^p$ norm... | Such inequalities appear all over the place in PDE theory. They all can be seen as instances of Ehrling's lemma. Here, you have
$$ (W^{2,p}_0(\Omega), ||\;||_3) \hookrightarrow (W^{1,p}_0(\Omega), ||\;||_2) \hookrightarrow (L^p(\Omega), ||\;||_1) $$
where
$$ ||u||_3 = ||D^2u||_p, ||u||_2 = ||Du||_p, ||u||_1 = ||u||_p. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$ Expanding the equation out gives
$(3^{2n}\times3)+(2^n\times2^2) \equiv 0\pmod{7}$
Is this correct? I'm a little hazy on my index laws.
Not sure if this is what I need to do? Am I on the right track?
| Note that
$$3^{2n+1} = 3^{2n} \cdot 3^1 = 3 \cdot 9^n$$ and $$2^{n+2} = 4 \cdot 2^n$$
Note that $9^{3k} \equiv 1 \pmod{7}$ and $2^{3k} \equiv 1 \pmod{7}$.
If $n \equiv 0 \pmod{3}$, then $$3 \cdot 9^n + 4 \cdot 2^n \equiv (3+4) \pmod{7} \equiv 0 \pmod{7}$$
If $n \equiv 1 \pmod{3}$, then $$3 \cdot 9^n + 4 \cdot 2^n \equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233937",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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infinitely many primes p which are not congruent to $-1$ modulo $19$. While trying to solve answer a question, I discovered one that I felt to be remarkably similar. The question I found is 'Argue that there are infinitely many primes $p$ that ar enot congruent to $1$ modulo $5$. I believe this has been proven. (bri... | Let $p_1,p_2,\dots,p_n$ be any collection of odd primes, and let $n=19p_1p_2\cdots p_n+2$. A prime divisor of $n$ cannot be one of the $p_i$. And $n$ has at least one prime divisor which is not congruent to $-1$ modulo $19$, else we would have $n\equiv \pm 1\pmod{19}$.
Remark: Not congruent is generally far easier to d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prime $p$ with $p^2+8$ prime I need to prove that there is only one $p$ prime number such that $p^2+8$ is prime and find that prime.
Anyway, I just guessed and the answer is 3 but how do I prove that?
| Any number can be written as $6c,6c\pm1,6c\pm2=2(3c\pm1),6c+3=3(2c+1)$
Clearly, $6c,6c\pm2,6c+3$ can not be prime for $c\ge 1$
Any prime $>3$ can be written as $6a\pm 1$ where $a\ge 1$
So, $p^2+8=(6a\pm 1)^2+8=3(12a^2\pm4a+3)$.
Then , $p^2+8>3$ is divisible by 3,hence is not prime.
So, the only prime is $3$.
Any numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which kind product of non-zero number non-zero cardinal numbers yields zero? Let $I$ be a non-empty set. $\kappa_i$ is non-zero cardinal number for all $i \in I$.
If without AC, then $\prod_{i \in I}\kappa_i=0$ seems can be true(despite I still cannot believe it).
But what property should $I$ and $\kappa_i$ have?
Can $... | The question is based on presuppositions that might not be true in the absence of AC. Let's consider the simplest non-trivial case, the product of countably many copies of 2, that is, $\prod_{n\in\mathbb N}\kappa_n$ where $\kappa_n=2$ for all $n$. A reasonable way to define this product would be: Take a sequence of s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$. Does the following limit exist? What is the value of it if it exists?
$$\lim\limits_{x\to\infty}f(x)^{1/x}$$
where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$ and $\{a_k\}\subset\mathbb{N}$ satisfies $a_k<a_{k+... | This limit does not exist in general. First observe that for any polynomial $P$ with non-negative coefficients we have
$$ \lim_{x\to\infty} P(x)^{1/x} = 1$$
and
$$ \lim_{x\to\infty} (e^x - P(x))^{1/x} = \lim_{x\to\infty} e (1-e^{-x}P(x))^{1/x} = e.$$
For ease of notation let
$$ e_n(x) = \sum_{k=n}^\infty \frac{x^k}{k!}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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lagrange multiplier with interval constraint Given a function $g(x,y,z)$ we need to maximize it given constraints $a<x<b, a<y<b$.
If the constraints were given as a function $f(x,y,z)$ the following equation could be used.
$\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$
How would I set up the initial equation given an i... | Maximize $g$ ignoring the constraint. If the solution fulfills the constraint, you're done. If not, there's no maximum, since it would have to lie on the boundary, but the boundary is excluded by the constraint.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/234303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does taking $\nabla\times$ infinity times from an arbitrary vector exists? Is it possible to get the value of:
\begin{equation}
\underbrace{\left[\nabla\times\left[\nabla\times\left[\ldots\nabla\times\right.\right.\right.}_{\infty\text{-times taking curl operator}}\mathbf{V}\left.\left.\left.\right]\right]\ldots\right]... | Two applications of $\nabla$ yield $\nabla \times (\nabla \times F) = -\nabla^2 F + \nabla(\nabla \cdot F)$. Why? Well, setting $F = \sum_i F_i e_i$ where $e_i$ is the standard cartesian frame of $\mathbb{R}^3$ allows the formula:
$$ (\nabla \times F)_k = \sum_{ij} \epsilon_{ijk} \partial_i F_j $$
Curling once more,
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Exponential operator on a Hilbert space Let $T$ be a linear operator from $H$ to itself. If we define $\exp(T)=\sum_{n=0}^\infty \frac{T^n}{n!}$ then how do we prove the function $f(\lambda)=exp(\lambda T)$ for $\lambda\in\mathbb{C}$ is differentiable on a Hilbert space?
| $$\frac{f(\lambda)-f(0)}{\lambda}=\frac{\exp(\lambda T)-Id}{\lambda} = \frac1\lambda\left( \sum_{n=1}^{\infty} \frac{\lambda^nT^n}{n!} \right) =
\sum_{n=1}^{\infty} \frac{\lambda^{n-1}T^n}{n!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/234418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Describing A Congruence Class The question is, "Give a description of each of the congruence classes modulo 6."
Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The relation would then be $R=\{(x,y)|x \equiv y~(mod~6)\}$
Then, $[n]_6 =\{x \in Z|~x \equiv ... | Let’s start with your correct description
$$[n]_6=\{x\in\Bbb Z:x\equiv n\!\!\!\pmod 6\}=\{x\in\Bbb Z:6\mid x-n\}$$
and actually calculate $[n]_6$ for some values of $n$.
*
*$[0]_6=\{x\in\Bbb Z:6\mid x-0\}=\{x\in\Bbb Z:6\mid x\}=\{x\in\Bbb Z:x=6k\text{ for some }k\in\Bbb Z\}$; this is just the set of all multiples o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Domain, codomain, and range This question isn't typically associated with the level of math that I'm about to talk about, but I'm asking it because I'm also doing a separate math class where these terms are relevant. I just want to make sure I understand them because I think I may end up getting answers wrong when I'm ... | $x=1$ is not in the domain because when $x=1$, $f(x)$ is undefined. And by definition, strictly speaking, a function defined on a domain $X$ maps every element in the domain to one and only element in the codomain.
The domain and codomain of a function depend upon the set on which $f$ is defined and the set to which el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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A limit $\lim_{n\rightarrow \infty}\sum_{k=1}^{n}\frac{k\sin\frac{k\pi}{n}}{1+(\cos\frac{k\pi}{n})^2}$ maybe relate to riemann sum Find
$$\lim_{n\rightarrow \infty}\sum_{k=1}^{n}\frac{k\sin\frac{k\pi}{n}}{1+(\cos\frac{k\pi}{n})^2}$$
I think this maybe relate to Riemann sum. but I can't deal with $k$ before $\sin$
| If there is no typo, then the answer is $\infty$. Indeed, let $m$ be any fixed positive integer and consider the final $m$ consecutive terms:
$$ \sum_{k=n-m}^{n-1} \frac{k \sin \frac{k \pi}{n}}{1 + \cos^2 \frac{k \pi}{n}}
= \sum_{k=1}^{m} \frac{(n-k) \sin \frac{k \pi}{n}}{1 + \cos^2 \frac{k \pi}{n}}. $$
As $n \to \inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Unimodular matrix definition? I'm a bit confused. Based on Wikipedia:
In mathematics, a unimodular matrix M is a square integer matrix
having determinant +1, 0 or −1. Equivalently, it is an integer matrix that is invertible over the integers.
So determinant could be +1, 0 or −1. But a matrix is invertible only if d... | Well spotted. In a case like this, it's a good idea to check the article's history (using the "View history" link at the top). In the present case, the error was introduced only two days ago by an anonymous user in this edit (which I just reverted).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What exactly does conjugation mean? In group theory, the mathematical definition for "conjugation" is:
$$
(g, h) \mapsto g h g^{-1}
$$
But what exactly does this mean, like in laymans terms?
| The following is equivalent to the second paragraph of Marc van Leeuwen's answer, but I think it might help emphasize how natural conjugation really is. With notation as in Marc's answer, let me write $h'$ for the conjugate $ghg^{-1}$. Then $h'$ is obtained by shifting $h$ along $g$ in the sense that, whenever $h$ se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Prove that: Every $\sigma$-finite measure is semifinite. I am trying to prove every $\sigma$-finite measure is semifinite. This is what I have tried:
Definition of $\sigma$-finiteness: Let $(X,\mathcal{M},\mu)$ is a measure space. Then, $ \mu$ is $\sigma$-finite if $X = \bigcup_{i=1}^{\infty}E_i$ where $E_i \in \math... | We can find $N$ such that $\mu\left(A\cap E_N\right)>0$ (otherwise, we would have for each $n$ that $\mu\left(A\cap\bigcup_{j=1}^nE_j\right)=0$ and $\mu\left(A\right)=\lim_{n\to +\infty}\mu\left(A\cap\bigcup_{j=1}^nE_j\right)$), and we have $\mu\left(A\cap E_N\right)\leqslant \mu\left( E_N\right)<+\infty$. Furthermore,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Probability of winning in the lottery In the lottery there are 5 numbers rolled from 35 numbers and for 3 right quessed numbers there is a third price.
What's the propability that we will win the third price if we buy one ticket with 5 numbers.
| Choose 5 from 35 in $\binom{35}{5}$ and from 5 numbers to get 3 exists $\binom{5}{3}=10$ possibilities and 2 other numbers you choose from 30 others thats not are in your ticket in $\binom{30}{2}=435 $ ways so total ways to win third place is $10\times435=4350$ ways.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/234956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. *help needed please* Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant.
This is what i've done so far....
Let $a,b,c$ be ... | The volume of a pyramid (of any shaped base) is $\frac13A_bh$, where $A_b$ is the area of the base and $h$ is the height (perpendicular distance from the base to the opposing vertex). In this particular case, we're considering a triangular pyramid, with the right triangle $OAB$ as a base and opposing vertex $C$. The ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/235041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Numbers to the Power of Zero I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as follows...
*
*Why does $n^{0}=1$ when $n\neq 0$? How does that get defined?
*What is $0^{0}$?... | To define $x^0$, we just cannot use the definition of repeated factors in multiplication. You have to understand how the laws of exponentiation work. We can define $x^0$ to be: $$x^0 = x^{n - n} = \frac{x^n}{x^n}.$$ Now, let us assume that $x^n = a$. It would then be simplified as $$\frac{x^n}{x^n} = \frac{a}{a} = 1.$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/235081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 10,
"answer_id": 7
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The inverse of the adjacency matrix of an undirected cycle Is there an expression for $A^{-1}$, where $A_{n \times n}$ is the adjacency matrix of an undirected cycle $C_n$, in terms of $A$?
I want this expression because I want to compute $A^{-1}$ without actually inverting $A$. As one answer suggests, $A$ is non-inve... | For $n=4$, the matrix in question is $$\pmatrix{0&1&0&1\cr1&0&1&0\cr0&1&0&1\cr1&0&1&0\cr}$$ which is patently noninvertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/235244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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$\epsilon$-$\delta$ proof involving differentiation in a defined neighborhood The problem states: Suppose $f'(b) = M$ and $M <0$. Find $\delta>0$ so that if $x\in (b-\delta, b)$, then $f(x) > f(b).$
This intuitively makes sense, but I am not exactly sure how to find $\delta$. I greatly appreciate any help I can receive... | Remember that the definition of derivative will imply that
$$
\lim_{x\to b^-}\frac{f(b)-f(x)}{b-x}=M.
$$
But, $M<0$ and $b-x>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/235319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a conformal map from the exterior of unit disk onto the exterior of an ellipse Find a conformal bijection $f(z):\mathbb{C}\setminus D\rightarrow \mathbb{C}\setminus E(a, b)$ where $E(a, b)$ is the ellipse $\{x + iy : \frac{x^2}{a}+\frac{y^2}{b}\leq1\}$
Here $D$ denotes the closed unit disk.
I hate to ask havi... | The conformal map $z\mapsto z+z^{-1}$ sends $\{|z|>R\}$ onto the exterior of ellipse with semi-axes $A=R+R^{-1}$ and $B=R-R^{-1}$. Note that $A^2-B^2=4$. Thus, you should multiply the given $a,b$ by a constant $C$ such that $(Ca)^2-(Cb)^2=4$, then solve $Ac=R+R^{-1}$ for $R$. After applying the map given above, the fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/235498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the limit of $x_1=0 , x_{n+1}=\frac{1}{1+x_n}$ I have had big problems finding the limit of the sequence $x_1=0 , x_{n+1}=\frac{1}{1+x_n}$. So far I've only succeeded in proving that for $n\geq2$: $x_n>0\Rightarrow x_{n+1}>0$
(Hopefully that much is correct: It is true for $n=2$, and for $x_{n+1}>0$ exactly whe... | It is obvious that $f:x\mapsto\frac1{1+x}$ is a monotonically decreasing continuous function $\mathbf R_{\geq0}\to\mathbf R_{\geq0}$, and it is easily computed that $\alpha=\frac{-1+\sqrt5}2\approx0.618$ is its only fixed point (solution of $f(x)=x$). So $f^2:x\mapsto f(f(x))$ is a monotonically increasing function tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Chain rule application I want to find $y'$ where $$ y = \frac{\frac{b}{a}}{1+ce^{-bt}}.$$ But I dont want to use quotient rule for differentiation. I want to use chain rule. My solution is:
Write $$y=\frac{b}{a}\cdot \frac{1}{1+ce^{-bt}}.$$ Then in $$\frac{1}{1+ce^{-bt}},$$ the inner function is $1+ce^{-bt}$ and the o... | Yes.
The outer function is $s\mapsto \displaystyle\frac1s$ or you can call its variable anything. And '$\cdot -bce^{-bt}$' should be in parenthesis: $\cdot (-bce^{-bt})$, else it seems correct.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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question about normal subgroups If $N$ is a normal subgroup of $G$ and $M$ is a normal subgroup of $G$, and if $MN=\{mn|m\in M,n\in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN$ is a normal subgroup of $G$.
The attempt:
I tried just starting by showing that $MN$ is a subgroup of $G$. I said let $a=m_1 n_1$ ... | $$m_1n_3m_2^{-1}=m_1m_2^{-1}(m_2n_3m_2^{-1})\in MN$$
General Lemma: if $\,M,N\,$ are subgroups of $\,G\,$ , $\,MN\,$ is a subgroup iff $\,MN=NM\,$ .
In particular, if $\,M\triangleleft G\,$ or$\,N\triangleleft G\,$ ,then $\,MN=NM\,$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Disjoint Equivalence Why do equivalence classes, on a particular set, have to be disjoint? What's the intuition behind it? I'd appreciate your help
Thank you!
| The idea behind an equivalence relation is to generalize the notion of equality.
The idea behind the equality relation is that something is only equal to itself. So two distinct objects are not equal.
With equivalence relation, if so, we allow two things to be "almost equal" (namely equal where it count, and we don't c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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A question on $\liminf$ and $\limsup$ Let us take a sequence of functions $f_n(x)$. Then, when one writes $\sup_n f_n$, I understand what it means: supremum is equal to upper bound of the functions $f_n(x)$ at every $x$. Infimum is defined similarly. Then when one writes $\lim \sup f_n$, then I understand following: Th... | 1 For any $ x $ there are $ n_{k(x)} $ such that
\begin{equation}
\limsup f_n(x) = \lim f_{n_{k(x)}}(x)
\end{equation}
2 Maybe $ f_n(x) $ can not converges. But, there are subindices $ n_k $ such that $ f_{n_k}(x) $ converges.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Finding the norm of the operators How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$?
1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot x(n).$$
What my thoughts were to use Banach-Steinhaus theorem but it seems straight forward a... | *
*As for each $n$, $|z(n)x(n)|^p\leqslant \lVert a\rVert_{\infty}|x(n)|^p$, then we certainly have $\lVert T_z\rVert\geqslant \lVert a\rVert_{\infty}$. To get the other inequality, fix $\delta$ and pick $k$ such that $|a(k)|\geqslant \lVert a\rVert_{\infty}-\delta$ (the case $a=0$ is obvious).
*We assume $t_j$ dist... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Summation over exponent $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$ Why does $$\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}{3}$$where does that 3 comes from?
Ok, from your answers I looked it up on Wikipedia Geometric Progression, but to derive the formula it says to multiply by $(1-r)$ not $(r-1)$ why is this case different?
| Here is an easy mnemonic. If you have a geometric sum, then
$$\sum {\rm geometric} = {{\rm first} - {\rm last}\over {1 - {\rm common \ Ratio}}}.$$
In this case, first is the first term, blast is the one beyond the last, and commonRatio is the common ratio of the terms. If the sum is finite and ${\rm commonRatio} > 1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example of a general random variable with finite mean but infinite variance Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, positive value, but $E(X^2)$ diverges.
| An example is a random variable $X$ having a student-t distribution with $\nu = 2$ degrees of freedom
Its mean is $E[X] = 0$ for $\nu > 1$, but its second moment $E[X^2] = Var[X] = \infty$ for $1 < \nu \le 2$
Edit: Oh wait, finite positive. Well, $X+1$, I guess.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/236181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Question on a proof of a sequence
I have some questions
1) In the forward direction of the proof, it employs the inequality $|x_{k,i} - a_i| \leq (\sum_{j=1}^{n} |x_{k,j} - a_j|^2)^{\frac{1}{2}}$. What exactly is this inequality?
2) In the backwards direction they claim to use the inequality $\epsilon/n$. I thought th... | $\def\abs#1{\left|#1\right|}$(1) We have that
$$ \abs{x_{k,i} - a_i}^2 \le \sum_{j=1}^n \abs{x_{k,j} - a_j}^2 $$
for sure as adding positive numbers makes the expression bigger. Now, exploiting the monotonicity of $\sqrt{\cdot}$, we have
$$ \abs{x_{k,i} - a_i} = \left(\abs{x_{k,i} - a_i}^2\right)^{1/2} \le \left(\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/236268",
"timestamp": "2023-03-29T00:00:00",
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Are there five complex numbers satisfying the following equalities? Can anyone help on the following question?
Are there five complex numbers $z_{1}$, $z_{2}$ , $z_{3}$ , $z_{4}$
and $z_{5}$ with $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|+\left|z_{5}\right|=1$
such that the smallest a... | Suppose you have solutions and express $z_i$ as $r_i e^{\theta_i}$.
(I use $s_i = \sin( \theta_i )$ and $c_i = \sin( \theta_i )$ to make notations shorter)
Then$$\begin{align*}
|z_i| + |z_j| - |z_i + z_j| & = |r_i e^{\theta_i}| + |r_j e^{\theta_j}| - |r_i e^{\theta_i} + r_j e^{\theta_j}| \\
& = r_i + r_j - |r_i(c_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/236313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Coin game - applying Kelly criterion I'm looking at a simple coin game where I have \$100, variable betting allowed, and 100 flips of a fair coin where H=2x stake+original stake, T=lose stake.
*
*If I'm asked to maximise the expected final net worth $N$, am I meant to simply bet a fraction of $\frac{1}{4}$ (accordin... | The Wikipedia essay says bet $p-(q/b)$, where $p$ is the probability of winning, $q=1-p$ of losing, and $b$ is the payment (not counting the dollar you bet) on a one dollar bet. For your game, $p=q=1/2$ and $b=2$ so, yes, bet one-fourth of your current bankroll.
Sorry, I'm not up to thinking about the logarithmic ques... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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how many ways can the letters in ARRANGEMENT can be arranged Using all the letters of the word ARRANGEMENT how many different words using all letters at a time can be made such that both A, both E, both R both N occur together .
| "ARRANGEMENT" is an eleven-letter word.
If there were no repeating letters, the answer would simply be $11!=39916800$.
However, since there are repeating letters, we have to divide to remove the duplicates accordingly.
There are 2 As, 2 Rs, 2 Ns, 2 Es
Therefore, there are $\frac{11!}{2!\cdot2!\cdot2!\cdot2!}=2494800$ w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What do I use to find the image and kernel of a given matrix? I had a couple of questions about a matrix problem. What I'm given is:
Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \vec{x} )=A\vec{x}$, where
$$A = \left(\begin{array}{crc}
1 & 2 & 2 & -5 & 6\\
-1 & -2 & -1 & 1 & ... | I could give an explanation for the most appreciated answer why image is calculated in this way. Image of a matrix is basically all the vectors you can obtain after this linear transformation. Let's say $A$ is a $2 \times 2$ matrix
$$A=\pmatrix {a_1 & b_1\\ a_2 & b_2}$$
. If we apply A as a linear transformation to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/236541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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What does brace below the equation mean? An example of what I am trying to understand is found on this page, at Eq. 3.
There are two braces under the equation... What is the definition of the brace(s) and how does it relate to Sp(t) and S[k]? This is what 4 years of calculus gets you 20+ years later...
http://en.wiki... | it is a shortcut to let you know that the expression above is equal to it (either by definition or by calculation)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/236621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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No Nonzero multiplication operator is compact Let $f,g \in L^2[0,1]$, multiplication operator $M_g:L^2[0,1] \rightarrow L^2[0,1]$ is defined by $M_g(f(x))=g(x)f(x)$. Would you help me to prove that no nonzero multiplication operator on $L^2[0,1]$ is compact. Thanks.
| We show that if $g$ is not the equivalence class of the null function, then $M_g$ is not compact. Let $c>0$ such that $\lambda(\{x,|g(x)|>c\})>0$ (such a $c$ exists by assumption). Let $S:=\{x,|g(x)|>c\}$, $H_1:=L^2[0,1]$, $H_2:=\{f\in H_1, f=f\chi_S\}$. Then $T\colon H_2\to H_2$ given by $T(f)=T_g(f)$ is onto. Indeed,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Convergence of $\sum_{n=1}^\infty \frac{a_n}{n}$ with $\lim(a_n)=0$. Is it true that if $(a_n)_{n=1}^\infty$ is any sequence of positive real numbers such that $$\lim_{n\to\infty}(a_n)=0$$
then,
$$\sum_{n=1}^\infty \frac{a_n}{n}$$
converges?
If yes, how to prove it?
| It is false. For $n\gt 1$, let $a_n=\dfrac{1}{\log n}$.
The divergence can be shown by noting that $\int_2^\infty \frac{dx}{x\log x}$ diverges. (An antiderivative is $\log\log x$.)
| {
"language": "en",
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How to show that two equivalence classes are either equal or have an empty intersection? For $x \in X$, let $[x]$ be the set $[x] = \{a \in X | \ x \sim a\}$.
Show that given two elements $x,y \in X$, either
a) $[x]=[y]$ or
b) $[x] \cap [y] = \varnothing$.
How I started it is, if $[x] \cap [y]$ is not empty, then $[x... | The problem with your "start" is that you are assuming exactly what you want to prove.
You need to apply what you know about the properties of an equivalence relation, in this case, denoted by $\;\sim\;$ You'll need to use the definitions of $[x], [y]$: $$[x] = \{a \in X | \ x \sim a\} \text{ and}\;\;[y] = \{a \in X... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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RSA solving for primes p and q knowing n = pq and p - q I was also given these:
$p+q=n-\phi(n)+1$
$p-q=\sqrt((p+q)^2-4n)$
$\phi(n)=(p-1)(q-1)$
$p>q$
I've been trying to manipulate this as a system of equations, but it's just not working out for me. I found a similar problem on this site, but instead of $pq$ and $p-q$... | We have
$$(p+q)^2=(p-q)^2+4pq.$$
Calculating the right-hand side is very cheap. Then calculating $p+q$ is cheap, a mild variant of Newton's Method. Once we know $p+q$ and $p-q$, calculating $p$ and $q$ is very cheap.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/236916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cancellation laws for function composition Okay I was asked to make a conjecture about cancellation laws for function composition. I figured it would go something like "For all sets $A$ and functions $g: A \rightarrow B$ and $h: A \rightarrow B$, $f \circ g = f \circ h$ implies that $g=h$."
I'm pretty sure $g=h$ isn't... | Notice that if there are distinct $b_1,b_2\in B$ such that $f(b_1)=f(b_2)$, you won’t necessarily be able to cancel $f$: there might be some $a\in A$ such that $g(a)=b_1$ and $h(a)=b_2$, but you’d still have $(f\circ g)(a)=(f\circ h)(a)$. Thus, you want $f$ to be injective (one-to-one). Can you prove that that’s suffic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Calculate a point on the line at a specific distance . I have two points which make a line $l$ , lets say $(x_1,y_1) , (x_2,y_2)$ . I want a new point $(x_3,y_3)$ on the line $l$ at a distance $d$ from $(x_2,y_2)$ in the direction away from $(x_1,y_1)$ . How should i do this in one or two equation .
| A point $(x,y)$ is on the line between $(x_1,y_1)$ and $(x_2,y_2)$ if and only if, for some $t\in\mathbb{R}$,
$$(x,y)=t(x_1,y_1)+(1-t)(x_2,y_2)=(tx_1+(1-t)x_2,ty_1+(1-t)y_2)$$
You need to solve
$$\begin{align*}d&=\|(x_2,y_2)-(tx_1+(1-t)x_2,ty_1+(1-t)y_2)\|=\sqrt{(tx_2-tx_1)^2+(ty_2-ty_1)^2}\\
&=\sqrt{t^2}\sqrt{(x_2-x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/237090",
"timestamp": "2023-03-29T00:00:00",
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Existence of two solutions I am having a problem with the following exercise.
I need to show the $x^2 = \cos x $ has two solutions.
Thank you in advance.
| Let $f(x)=x^2-\cos x$. Note that the curve $y=f(x)$ is symmetric about the $y$-axis. It will thus be enough to show that $f(x)=0$ has a unique positive solution. That there is a unique negative solution follows by symmetry.
There is a positive solution, since $f(0)\lt 0$ and $f(100)\gt 0$. (Then use the Intermediate V... | {
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How do I get the residue of the given function? I'm reading the solution of the integral:
$$\int\limits_{-\infty}^{\infty} dx\frac{e^{ax}}{1+e^x}$$
by the residue method. And I understood everything, but how to get the residue of $\frac{e^{az}}{1+e^z}$ (the book just states that the residue is $-e^{i\pi a}$).
I know th... | $$\lim_{z\to\pi i}(z-\pi i)\frac{e^{az}}{1+e^z}\stackrel{\text{L'Hopital}} = \lim_{z\to\pi i}\frac{e^{az}}{e^z} = -e^{a\pi i}$$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the set of all real numbers uncountable? I understand Cantor's diagonal argument, but it just doesn't seem to jive for some reason.
Lets say I assign the following numbers ad infinitum...
*
*$1\to 0.1$
*$2\to 0.2$
*$3\to 0.3$
...
*$10\to 0.10$
*$11\to 0.11$
and so on... How come there's supposedly at le... | How come there's supposedly at least one more real number than you can map to a member of $\mathbb N$?
Well, suppose there isn't - that Cantor's conclusion, his theorem, is wrong, because our enumeration covers all real numbers. Wonderful, but let us see what happens when we take our enumeration and apply Cantor's dia... | {
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Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices? I have this problem from my Graphics course. Given this transformation matrix:
$$\begin{pmatrix}
-2 &-1& 2\\
-2 &1& -1\\
0 &0& 1\\
\end{pmatrix}$$
I need to extract translation, rotation and scale matrices.
I've ... | It appears you are working with Affine Transformation Matrices, which is also the case in the other answer you referenced, which is standard for working with 2D computer graphics. The only difference between the matrices here and those in the other answer is that yours use the square form, rather than a rectangular au... | {
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"answer_id": 1
} |
Limit:$ \lim\limits_{n\rightarrow\infty}\left ( n\bigl(1-\sqrt[n]{\ln(n)} \bigr) \right )$ I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$
what gives $$\lim_{n\rightarrow\in... | Use Taylor!
$$n(1-\sqrt[n]{\log n}) = n (1-e^{\frac{\log\log n}{n}}) \approx n\left(1-\left(1+\frac{\log\log n}{n}\right)\right) = - \log\log n$$
which clearly tends to $-\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/237446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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Prove sum is bounded I have the following sum:
$$
\sum\limits_{i=1}^n \binom{i}{i/2}p^\frac{i}{2}(1-p)^\frac{i}{2}
$$
where $p<\frac{1}{2}$
I need to prove that this sum is bounded. i.e. it doesn't go to infinity as n goes to infinity.
| And instead of an explicit bound, you may use Stirling's formula, which yields
$\displaystyle {n \choose n/2} \sim \sqrt{2 / \pi} \cdot n^{-1/2} 2^n$ as $n \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/237495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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is $0.\overline{99}$ the same as $\lim_{x \to 1} x$? So we had an interesting discussion the other day about 0.999... repeated to infinity, actually being equal to one. I understand the proof, but I'm wondering then if you had the function...
$$
f(x) = x* \frac{(x-1)}{(x-1)}
$$
so
$$
f(1) = NaN
$$
and
$$
\lim_{x \to ... | If $0.\overline9=1$ then $f(0.\overline9)$ is as undefined as $f(1)$ is. However indeed $\lim_{x\to 1}f(x)=1$ as you said.
The reason for the above is simple. If $a$ and $b$ are two terms, and $a=b$ then $f(a)=f(b)$, regardless to what $f$ is or what are the actual terms. Once you agreed that $0.\overline9=1$ we have t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/237571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
4 Points on Circumference of Circle and center This is actually a computer science question in that I need to create a program that will determine the center of a circle given $4$ points on its circumference.
Does anyone know the algorithm, theorem or method? I think it has something to do with cyclic quadrilaterals.
... | The perpendicular bisector of a chord of a circle goes through the center of the circle. Therefore, if you have two chords, then the perpendicular bisectors intersect at exactly the center of the circle. Here is a picture of what I'm describing.
So, given four points on the circle, draw chords between pairs of them, d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Generating function for binomial coefficients $\binom{2n+k}{n}$ with fixed $k$ Prove that
$$
\frac{1}{\sqrt{1-4t}} \left(\frac{1-\sqrt{1-4t}}{2t}\right)^k = \sum\limits_{n=0}^{\infty}\binom{2n+k}{n}t^n,
\quad
\forall k\in\mathbb{N}.
$$
I tried already by induction over $k$ but i have problems showing the statement hold... | Due to a recent comment on my other answer, I took a second look at this question and tried to apply a double generating function.
$$
\begin{align}
&\sum_{n=0}^\infty\sum_{k=-n}^\infty\binom{2n+k}{n}x^ny^k\\
&=\sum_{n=0}^\infty\sum_{k=n}^\infty\binom{k}{n}\frac{x^n}{y^{2n}}y^k\\
&=\sum_{n=0}^\infty\frac{x^n}{y^{2n}}\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/237810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 6,
"answer_id": 1
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Rolle's Theorem Let $f$ be a continuous function on $[a,b]$ and differentiable on $(a,b)$, where $a<b$. Suppose $f(a)=f(b)$. Prove that there exists number $c_{1},c_{2},...,c_{2012}$ $\in$ $(a,b)$ satisfying $c_{1} < c_{2} <...< c_{2012}$ and $f'(c_{1})+f'(c_{2})+...+f'(c_{2012})=0$.
I believe it has something to do wi... | Hint: Given $n\in\mathbb{N}$, consider the function $g_n(x)=\sum_{k=0}^{n-1}f(a+\frac{(b-a)(x+k)}{n})$, $x\in[0,1]$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 2
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Mathematics induction on inequality: $2^n \ge 3n^2 +5$ for $n\ge8$ I want to prove
$2^n \ge 3n^2 +5$--call this statement $S(n)$--for $n\ge8$
Basis step with $n = 8$, which $\text{LHS} \ge \text{RHS}$, and $S(8)$ is true. Then I proceed to inductive step by assuming $S(k)$ is true and so
$2^k \ge 3k^2 +5 $
Then $S(... | The missing step (because there is indeed a missing step) is that $2\cdot(3k^2+5)\geqslant3(k+1)^2+5$. This inequality is equivalent to $3k^2-6k+2\geqslant0$, which obviously holds for every $k\geqslant8$ since $3k^2-6k+2=3k(k-2)+2$, hence you are done.
The structure of the proof that $S(k)$ implies $S(k+1)$ for every ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$ I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into $\psi(0,\omega... | If you complete the square in the argument of the exponentials, $$ -\frac{1}{4}(t^2 + 4i \omega t) \to -\frac{1}{4}(t^2+4i\omega t -4\omega^2) -\omega^2 = -\frac{1}{4}(t+i2\omega)^2-\omega^2. $$ After a change of variables $u=\frac{t}{2}+i\omega$, the integral becomes $$2e^{-\omega^2}\int_{-\infty -i\omega}^{\infty -i\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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is there a large time behavoiur of (compound) Poisson processes similiar to Law of iterated logrithm for Brownian Motion Can someone please point me to a reference/answer me this question?
From the law of iterated logarithm, we see that Brownian motion with drift converge to $\infty$ or $-\infty$.
For a Poisson process... | Since $N_t/t\to\lambda$ almost surely, $X_t=\exp(aN_t-(e^a-1)\lambda t + rt)=\exp(\mu t+o(t))$ almost surely, with $\mu=(1+a-\mathrm e^a)\lambda+r$. If $\mu\ne0$, this yields that $X_t\to0$ or that $X_t\to+\infty$ almost surely, according to the sign of $\mu$.
If $\mu=0$, the central limit theorem indicates that $N_t=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.
(Please note that $e$ in the question is the... | given $g^2=e$ for all $g\in G$
So $g=g^{-1}$ for all $g\in G$
Let,$a,b\in G$
Now $ab=a^{-1}b^{-1}
=(ba)^{-1}
=ba$
So $ab=ba$ for all $a,b\in G$ .Hence $G$ is Abelian Group.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/238171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 14,
"answer_id": 4
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Basis for this $\mathbb{P}_3$ subspace. Just had an exam where the last question was:
Find a basis for the subset of $\mathbb{P}_3$ where $p(1) = 0$ for all $p$.
I answered $\{t,t^2-1,t^3-1\}$, but I'm not entirely confident in the answer. Did I think about the question in the wrong way?
| Another way to get to the answer:
$P_3=\{{ax^3+bx^2+cx+d:a,b,c,d{\rm\ in\ }{\bf R}\}}$. For $p(x)=ax^3+bx^2+cx+d$ in $P_3$, $p(1)=0$ is $$a+b+c+d=0$$ So, you have a "system" of one linear equation in 4 unknowns. Presumably, you have learned how to find a basis for the vector space of all solutions to such a system, or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the Cumulative Distribution Function of the following random variable? Suppose that we have $2n$ iid random variables $X_1,…,X_n,Y_1,…,Y_n$ where $n$ is a large number.
I want to find $P((k∑_iX_iY_i+(∑_iX_i)(∑_jY_j))<c)$ for any integer c.
Since $n$ is a large number and all the random variables are $iid$, usi... | $$Z = \sum_{i=1}^n \sum_{j=1}^n X_i Y_j = \left(\sum_{i=1}^n X_i\right)\left(\sum_{j=1}^n Y_j\right)$$
If $n$ is large, $S_X = \sum_i X_i$ and $S_Y = \sum_j Y_j$ are approximately normal. They have means $n\mu$ and standard deviations $\sqrt{n} \sigma$ where each $X_i$ and $Y_j$ have mean $\mu$ and standard deviation $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In the ring $\mathbb Z[i]$ explain why our four units $\pm1$ and $\pm i$ divide every $u\in\mathbb Z[i]$. In the ring $\mathbb Z[i]$ explain why our four units $\pm1$ and $\pm i$ divide every $u\in\mathbb Z[i]$.
This is obviously a elementary question, but Gaussian integers are relatively new to me. I found this exerc... | As important as it is to understand why the units divide everything, it's also important to understand why those are the only units (as they are probably being referred to as "the units" in class).
Let $z=a+bi$. Then we can define the norm of $z$ to be $N(z)=|z|^2=z\overline z = a^2+b^2$. Note, some people call $|z|$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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When is a factorial of a number equal to its triangular number? Consider the set of all natural numbers $n$ for which the following proposition is true.
$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$
Here's an example:
$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot 3=\prod_{k=1}^{3}k$$
Therefore, $3$ is in this set.
Does this... | The other answers are fine but it shouldn't be necessary to actually carry out the induction to see that the solution set is finite: the triangular numbers grow quadratically and factorials grow super-exponentially. A more interesting problem would be: how many solutions $(m,n)$ are there to $\sum_{k=1}^{m}{k} = \prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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numbers' pattern It is known that
$$\begin{array}{ccc}1+2&=&3 \\ 4+5+6 &=& 7+8 \\
9+10+11+12 &=& 13+14+15 \\\
16+17+18+19+20 &=& 21+22+23+24 \\\
25+26+27+28+29+30 &=& 31+32+33+34+35 \\\ldots&=&\ldots
\end{array}$$
There is something similar for square numbers:
$$\begin{array}{ccc}3^2+4^2&=&5^2 \\ 10^2+11^2+12^2 &=& 13... | Let's start by proving the basic sequences and then where and why trying to step it up to cubes fails. I don't prove anything just reduce the problem to a two variable quartic Diophantine equation.
Lemma $1 + 2 + 3 + 4 + \ldots + n = T_1(n) = \frac{n(n+1)}{2}$.
Corollary $(k+1) + (k+2) + \ldots + (k+n) = -T_1(k) + T_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
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About the asymptotic behaviour of $\sum_{n\in\mathbb{N}}\frac{x^{a_n}}{a_n!}$ Let $\{a_n\}_{n\in\mathbb{N}}$ be an increasing sequence of natural numbers, and
$$ f_A(x)=\sum_{n\in\mathbb{N}}\frac{x^{a_n}}{a_n!}. $$
There are some cases in which the limit
$$ l_A=\lim_{x\to+\infty} \frac{1}{x}\,\log(f_A(x)) $$
does not e... | It is true that $l_A=1$. The logic is similar to my answer to $\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$. Firstly, the terms after $n=3x$ don't matter:
$$\sum_{n=3x}^\infty x^n/n! <\sum_{n=3x}^\infty x^n /(3x/e)^n=C$$ by Stirling approximation.
But for $n<3x$ ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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Plotting an integral of a function in Octave I try to integrate a function and plot it in Octave.
Integration itself works, i.e. I can evaluate the function g like g(1.5) but plotting fails.
f = @(x) ( (1) .* and((0 < x),(x <= 1)) + (-1) .* and((1 <x),(x<=2)));
g = @(x) (quadcc(f,0,x));
x = -1.0:0.01:3.0;
plot(x,g(x))... | You could use cumtrapz instead of quadcc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/238697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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$Μ : K$ need not be radical Show that $Μ : K$ need not be radical, Where $L : K$ is a radical extension in $ℂ$ and $Μ$ is an intermediate field.
| Let $K = \Bbb Q$ and $M$ be the splitting field of $X^3 - 3X + 1 \in \Bbb Q[X]$.
$M$ can be embedded into $\Bbb R$, so it is not a radical extension by casus irreducibilis.
However, $X^3 - 3X + 1$ has a solvable Galois group $C_3$ (the cyclic group of order $3$), so $M$ can be embedded into some field $L$ that is radic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Tangent Spaces and Morphisms of Affine Varieties In page 205 of "Algebraic Curves, Algebraic Manifolds and Schemes" by Shokurov and Danilov, the tangent space $T_x X$ of an affine variety $X$ at a point $x \in X$ is defined as the subspace of $K^n$, where $K$ is the underlying field, such that $\xi \in T_x X$ if $(d_x ... | If $X\subset K^n$ and $Y\subset K^m$ are affine subvarieties , the map $f:X\to Y$ is the restriction of some polynomial map $F:K^n\to K^m: x\mapsto (F_1(x),...,F_m(x))$, where the $F_i$'s are polynomials $F_i\in K[T_1,...,T_n]$.
The map $d_x f : T_x X \rightarrow T_{f(x)} Y$ is the restriction to $T_x(X)$ of the li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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proof: set countable iff there is a bijection In class we had the following definiton of a countable set:
A set $M$ is countable if there is a bijection between $\mathbb N$ and $M$.
In our exam today, we had the following thesis given:If $A$ is a countable set, then there is a bijection $\mathbb N\rightarrow A$.
So I ... | Suppose that $A$ is countable by your definition; then there is a bijection $f:\Bbb N\to A$. Because $f$ is a bijection, $f^{-1}$ is also a bijection, so it’s the desired bijection from $A$ to $\Bbb N$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/238934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Exponents in Odd and Even Functions I was hoping someone could show or explain why it is that a function of the form $f(x) = ax^d + bx^e + cx^g+\cdots $ going on for some arbitrary length will be an odd function assuming $d, e, g$ and so on are all odd numbers, and likewise why it will be even if $d, e, g$ and so on a... | If the exponents are all odd, then $f(x)$ is the sum of odd functions, and hence is odd.
If the exponents are all even, then $f(x)$ is the sum of even functions, and hence is even.
As far as your last question, the sum of an odd function and even function is neither even nor odd.
Proof: Sum of Odd Functions is Odd:
Gi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Construction of an integrable function with a function in $L^2$ I have this really simple question, but I cannot figure out the answer. Suppose that $f\in L^2([0,1])$. Is it true that $f/x^5$ will be in $L^1([0,1])$?
Thanks!
Edit: I was interested in $f/x^{1/5}$.
| The statement
$$f \in L^2([0,1])\Rightarrow fx^{-\frac{1}{5}} \in L^1([0,1])$$
is true. It is an easy application of Hölder inequality. Infact, by hypothesis $f\in L^2([0,1])$; on the other hand, we have $g(x)=x^{-1/5}\in L^2([0,1])$ so by Hölder (indeed, this is the case "Cauchy-Schwarz")
$$
\Vert fg \Vert_1 \le \V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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The dimension of the component of a variety Mumford claimed the following result:
If $X$ is an $r$-dimensional variety and $f_{1},...,f_{k}$ are polynomial functions on $X$. Then every component of $X\cap V(f_{1},...,f_{k})$ has dimension $\ge r-k$.
He suggested this is a simple corollary of the following result:
Assu... | To deduce the corollary, let $Y$ be $k$-dimensional affine space, and let $\phi$ be the map sending a point $x \in X$ to $(f_1(x), f_2(x), \ldots, f_k(x))$. Then $X \cap V(f_1, \ldots, f_k)$ is the preimage $\phi^{-1}(0,0,\ldots,0)$, hence, by the result, its dimension is at least $r-k$.
(The fact that $\phi$ might no... | {
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"url": "https://math.stackexchange.com/questions/239122",
"timestamp": "2023-03-29T00:00:00",
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Showing that $|||A-B|||\geq \frac{1}{|||A^{-1}|||}$? $A,B\in M_n$, $A$ is non-singular and $B$ is singular. $|||\cdot|||$ is any matrix norm on $M_n$, how to show that $|||A-B||| \geq \frac{1}{|||A^{-1}|||}$?
The hint is let $B=A[I-A^{-1}(A-B)]$, but I don't know how to use it.
Appreciate any help!
update: is $\geq$,no... | Let's sharpen the hint to $A^{-1}B = I - A^{-1}(A-B)$. First you should check that this identity is correct.
Now pick any $v$ such that $Bv = 0$ and $\|v \| = 1$. By assumption, such a $v$ exists. Apply both sides of the identity, play around with it, take norms, see if you can get something that resembles the statemen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basis, dense subset and an inequality Let $V \subset H$, where $V$ is separable in the Hilbert space $H$.
So there is a basis $w_i$ in $V$ such that, for each $m$, $w_1, ..., w_m$ are linearly independent and the finite linear combinations are dense in $V$.
Let $y \in H$, and define $y_m = \sum_{i=1}^m a_{im}w_i$ such ... | It is not true. Choose $y=0$ and $a_{1,m} = \frac{1}{m}$. Then $y_m \to y$, but it is never the case that $\|y_m\| \leq C \|y\|$.
Elaboration:
This is because $\|y_m\| = \frac{1}{m} \|w_1\|$, the $w_i$ are linearly independent, hence non-zero. Hence $\|y_m\| = \frac{1}{m} \|w_1\| > 0$ for all $m$. There is no choice of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tate $p$-nilpotent theorem Tate $p$-nilpotent Theorem. If $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$ such that $P \cap N \leq \Phi (P)$, then $N$ is $p$-nilpotent.
My question is the following:
If $P \cap N \leq \Phi (P)$ for only one Sylow p-subgroup of $G$, is $N$ $p$-nilpotent?
Remark: $G... | That situation is not possible. Let $P$ be a Sylow $p$-subgroup such that $P \cap N \leqslant \Phi(P)$ and consider $Q\cap N$ for another Sylow $p$-subgroup $Q$. We have that there is a $g$ so that $P^g=Q$, and since $N$ is normal, $$(P\cap N)^g=P^g\cap N^g=P^g \cap N=Q\cap N\leq \Phi(P)^g=\Phi(P^g)=\Phi(Q).$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve the recurrence relation:$ T(n) = \sqrt{n} T \left(\sqrt n \right) + n$ $$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$$
Master method does not apply here. Recursion tree goes a long way. Iteration method would be preferable.
The answer is $Θ (n \log \log n)$.
Can anyone arrive at the solution.
| Let $n = m^{2^k}$. We then get that
$$T(m^{2^k}) = m^{2^{k-1}} T (m^{2^{k-1}}) + m^{2^{k}}$$
\begin{align}
f_m(k) & = m^{2^{k-1}} f_m(k-1) + m^{2^k} = m^{2^{k-1}}(m^{2^{k-2}} f_m(k-2) + m^{2^{k-1}}) + m^{2^k}\\
& = 2 m^{2^k} + m^{3 \cdot 2^{k-2}} f_m(k-2)
\end{align}
$$m^{3 \cdot 2^{k-2}} f_m(k-2) = m^{3 \cdot 2^{k-2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 0
} |
Straightening the boundary in concrete examples Let $\Omega \subset \mathbb{R}^d$ be open and with $C^1$ boundary $\Gamma$.
For any given point $x_0 \in \Gamma$ we know there's a neighborhood where
$\Gamma$ is the graph of some $C^1$ function $\gamma : \mathbb{R}^{d - 1}
\longrightarrow \mathbb{R}^d, x' \longmapsto \ga... | If $\phi(x')$ denotes the $d$-th component of the normal vector at $(x',\gamma(x'))$, then first of all it is immediate from the graph structure that $\phi(x') \ne 0$. Let $S(x',y_d) = (x',\phi(x')y_d)$, and write $\tilde{T} = S \circ T$. Then $\tilde{T}$ is a $\mathcal{C}^1$ diffeomorphism which straightens the bounda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subset of a finite set is finite We define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$.
How can we prove that a subset of a finite set is finite? It is of course sufficient to show that for a subset of $\{0,\ldots,n-1\}$. But how do I do t... | I have run into this old question and was surprised that noone seemed to have said the following.
The actual definition of finite set is the following: A set $A$ is finite if every injection $A\rightarrow A$ is a bijection.
(Note: this definition does not require the set $\mathbb{N}$)
Now let $B$ a finite set and $A\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 7,
"answer_id": 6
} |
Why is $\operatorname{Var}(X+Y) = \operatorname{Cov}(X,X) + \operatorname{Cov}(X,Y) + \operatorname{Cov}(Y,X) + \operatorname{Cov}(Y,Y)$ I know $\operatorname{Cov}(X,Y) = E[(X-u_x)(Y-u_y)]$ and
$$
\operatorname{Cov}(X+Y, Z+W) = \operatorname{Cov}(X,Z) + \operatorname{Cov}(X,W) + \operatorname{Cov}(Y,Z) + \operatorname{... | A quick way: Note from the definition of variance that $\text{Var}(T)=\text{Cov}(T,T)$. Now in your formula for $\text{Cov}(X+Y, Z+W)$, set $Z=X$ and $Y=W$. You will get exactly the formula you want to derive.
A slow way: We can work with just your basic defining formula for covariance. Note that
$$\text{Var}(X+Y)=E... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$G=\langle a,b\mid aba=b^2,bab=a^2\rangle$ is not metabelian of order $24$ This is my self-study exercise:
Let $G=\langle a,b\mid aba=b^2,bab=a^2\rangle$. Show that $G$ is not metabelian.
I know; I have to show that $G'$ is not an abelian subgroup. The index of $G'$ in $G$ is 3 and doing Todd-Coxeter Algorithm for f... | $abab=a^3=b^3$, so $Z := \langle a^3 \rangle$ is central. Modulo $Z$, we get the standard presentation $\langle a,b \mid a^3, b^3, (ab)^3 \rangle$ of $A_4$. Also, module $G'$, we have $a^2=b$, $b^2=a$, so $a^3=1$, and hence $Z \le G'$. Also, $ab,ba \in G'$ and $abba = a^2ba^2=bab^3ab=baabb^3$, so $G'$ is not abelian pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A Book for abstract Algebra I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined to have multiplicative identities. Also modules are defined slightly differently....etc) Gi... | There is a less famous but very nice book Abstract Algebra by Paul B. Garrett
and then there is the old book "A survey of modern algebra" by Birkhoff
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/239734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 10,
"answer_id": 8
} |
$\int 2^x \ln(x)\, \mathrm{d}x$ I found this problem by a typo. My homework problem was $\int 2^x \ln(2) \, \mathrm{d}x$ which is $2^x + C$ by the Fundamental Thm of Calculus. I want to be able to solve what I wrote down incorrectly in my homework.
What I wrote for my homework is $\int 2^x \ln(x)\, \mathrm{d}x$ and W... | First you did a mistake here:
$$\int 2^x \ln(x)\, \mathrm{d}x = \frac{\ln(x)2^x}{\ln 2} - \left[ \frac{2^x \ln x}{\ln 2} - \int \ln(x) 2^x\, \mathrm{d}x \right]\Rightarrow \frac{\ln(x)2^x}{\ln 2} -\frac{2^x \ln x}{\ln 2}= 0$$
You can't just cancel the integrals, as you will lose the constant of integration. For example... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the sum of the first $n$ terms of $\sum^n_{k=1}k^3$ The question:
Find the sum of the first $n$ terms of $$\sum^n_{k=1}k^3$$
[Hint: consider $(k+1)^4-k^4$]
[Answer: $\frac{1}{4}n^2(n+1)^2$]
My solution:
$$\begin{align}
\sum^n_{k=1}k^3&=1^3+2^3+3^3+4^3+\cdots+(n-1)^3+n^3\\
&=\frac{n}{2}[\text{first term} + \text{la... | For a geometric solution, you can see theorem 3 on the last page of this PDF. Sorry I did not have time to type it here. This solution was published by Abu Bekr Mohammad ibn Alhusain Alkarachi in about A.D. 1010 (Exercise 40 of appendix E, page A38 of Stewart Calculus 5th edition).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/239909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Dual of $\ell_\infty(X)$ Given a Banach space $X$. Consider the space $\ell_\infty(X)$ which is the $\ell_\infty$-sum of countably many copies of $X$. Is there any accessible respresentation of the dual space $\ell_\infty(X)^*$? In particular, is this dual space isomorphic to the space of finitely additive $X^*$-valued... | There is no good description of the dual of $\ell_\infty(X)$ as far as I know.
If $X$ is finite dimensional, then the answer to your second question is yes.
Otherwise, it is no, for there is no way to define an action of a finitely additive $X^*$-valued measure on $\ell_\infty(X)$ if the ball of $X$ is not compact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/240072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
quadratic and bilinear forms Does a quadratic form always come from symmetric bilinear form ?
We know when $q(x)=b(x,x)$ where $q$ is a quadratic form and $b$ is a symmetric bilinear form.
But when we just take a bilinear form and $b(x,y)$ and write $x$ instead of $y$,does it give us a quadratic form ?
| If we have b symmetric bilinear form we can get q quadratic form
$q\colon V \to \mathbb{R}$
q(v)=b(v,v)
conversely if q is a quadratic form
$q\colon V \to \mathbb{R}$
we can define
$\frac 12$(q(v+w)-q(v)-q(w)):=b(v,w)
the vital answer is you just get a bilinear form not always a symmetric bilinear
form.
because the def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
$L_1$ norm of Gaussian random variable Ok. This is bit confusing. Let $g$ be a Gaussian random variable (normalized, i.e. with mean 0 and standard deviation 1). Then, in the expression $$\|g\|_{L_1}=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}|x|\exp(\frac{-x^2}{2})dx =\sqrt{\frac{2}{\pi}},$$ shouldn't the term $|x|$ not app... | If $X$ is a random variable with density $f$, and $\phi$ is a measurable function, then $E[\phi(X)]=\int_{\Bbb R}\phi(t)f(t)dt$. As the $L^1$ norm of a random variable $X$ is $E[|X|]$, we have, when $X$ is normally distributed, the announced result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/240202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Method of Undetermined Coeffecients - how to assume the form of third degree equation. An example differential equations questions asks me to solve;
$$y''' - 2y'' -4y'+8y = 6xe^{2x}$$
I begun by solving the homogeneous equation with $m^3 - 2m^2 -4m+8 =0$ and getting the answer
$$y(x) = c_1e^{2x} + c_2xe^{2x}+c_3e^{-2x... | Your equation is $y'''-2y''-4y'+8y=6xe^{2x}$. Now change the $y'$ to $Dy$ form as follows. So $$y'''\to D^3y,\\ y''\to D^2y, \; \; \text{and} \;\;y'\to Dy,$$ So by new arranging respect to $D$ operator we get our equation as: $$D^3y-2D^2y-4Dy+8y=6xe^{2x}$$ or by factoring and expanding $$(D^3-2D^2-4D+8)y=(D-2)^2(D+2)y=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Line integral vs Arc Length I am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so:
$s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d} x$ for the arc length
and here's a line integral equation:
$\int_c fds=\int_a^b f(r(t))\cdot ... | Suppose you have a curve $C$ parametrized as $\mathbf{g}(t)$ for $0\le t\le 1$. Then the arc length of $C$ is defined as
$$\int_0^1\|\mathbf{g}'(t)\|\ dt$$
An intuitive way of think of the above integral is to interpret the derivative $\mathbf{g}'$ as velocity. Then the above integral is basically the statement that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
"answer_id": 0
} |
Show that if m/n is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better
Claim: If $m/n$ is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better.
My attempt at the proof:
Let d be the distance between $\sqrt{2}$ and some estimate, s.
So we have $d=s-\sqrt{2}$
Define $d'=m/n-\sqrt{2}$ and $d'... | Assume $\dfrac mn\ne\sqrt2;$ otherwise $\dfrac mn$ is $\sqrt2$, not an approximation.
Then $d'\ne0$ so we can compute $\dfrac {d''}{d'}=\dfrac{\dfrac{m+2n}{m+n}-\sqrt2}{\dfrac mn-\sqrt2}=
\dfrac n{m+n}\dfrac{m+2n-\sqrt2(m+n)}{m-\sqrt2n}$
$=\dfrac n{m+n}\dfrac{m-\sqrt2n-\sqrt2(m-\sqrt2n)}{m-\sqrt2n}=\dfrac {1}{1+\dfrac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Radius, diameter and center of graph
The eccentricity $ecc(v)$ of $v$ in $G$ is the greatest distance from $v$ to any other node.
The radius $rad(G)$ of $G$ is the value of the smallest eccentricity.
The diameter $diam(G)$ of $G$ is the value of the greatest eccentricity.
The center of $G$ is the set of nodes $v$ such... | I think for the path graph Pn, the diameter is n−1.But the radius is (n-1/2)rounded up to the nearest integer.
For example,P3 it has radius of 1 but not 2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/240556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 1
} |
How to finish proof that $T$ has an infinite model? I'm trying to prove the following: If $T$ is a first-order theory with the property that for every natural number $n$ there is a natural number $m>n$ such that $T$ has an $m$-element model then $T$ has an infinite model.
My thoughts: If $M$ is an $n$-element model the... | This is a standard fact. The result you are looking for is exactly the compactness theorem, but you can also do it directly. Just take an ultraproduct of a sequence $M_i$ of larger and larger finite models. Since every one of these models $T$, so does the ultraproduct, by Łoś's theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/240622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Recursive Integration over Piecewise Polynomials: Closed form? Is there a closed form to the following recursive integration?
$$
f_0(x) =
\begin{cases}
1/2 & |x|<1 \\
0 & |x|\geq1
\end{cases}
\\
f_n(x) = 2\int_{-1}^x(f_{n-1}(2t+1)-f_{n-1}(2t-1))\mathrm{d}t
$$
It's very clear that this converges against some function an... | Suppose $f$ is a fixed point of the iterations. Then
$$f(x) = 2\int_{-1}^x\big(f(2t+1)-f(2t-1)\big)\,\mathrm{d}t,$$
which, upon differentiating both sides by $x$, implies that
$$f'(x) = 2\big(f(2x+1)-f(2x-1)\big).$$
I'll assume that $f$ vanishes outside $[-1,1]$, which you can presumably prove from the initial conditio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 0
} |
Sufficient condition for differentiability at endpoints. Let $f:[a,b]\to \mathbb{R}$ be differentiable on $(a,b)$ with derivative $g=f^{\prime}$ there.
Assertion: If $\lim_{x\to b^{-}}g(x)$ exists and is a real number $\ell$ then $f$ is differentiable at $b$ and $f^{\prime}(b)=\ell$?
Is this assertion correct? If so p... | Since
$$
f(b+h)-f(b)=f'(\xi)h
$$
for some $\xi \in (b-h,h)$, you can let $h \to 0^{-}$ and conclude that $f'(b)=\lim_{x \to b-}f'(x)$.
On the other hand, consider $f(x)=x^2 \sin \frac{1}{x}$. It is easy to check that $\lim_{x \to 0} f'(x)$ does not exist, and yet $f'(0)=0$.
Edit: this answer assumes tacitly that $f$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/240757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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