Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Determine a holomorphic function by means of its values on $ \mathbb{N} $ This is exercise 5, page 236 from Remmert, Theory of complex functions
For each of the following properties produce a function which is holomorphic in a neighborhood of $ 0 $ or prove that no such function exists:
i) $ f (\frac{1}{n}) = (-1)^{n}\... | Your title is misleading, as you cannot determine a holomorphic function from its values on $\mathbb{N}$. However, in this case you can determine it, using the uniqueness theorem for analytic functions: if $f$ and $g$ are two analytic functions and there is a convergent series $a_n$ such that $f(a_n)=g(a_n)$ for all $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Number of pairs $(a, b)$ with $gcd(a, b) = 1$ Given $n\geq1$, how can I count the number of pairs $(a,b)$, $0\leq a,b \leq n$ such that $gcd(a,b)=1$?
I think the answer is $2\sum_{i}^{n}\phi(i)$. Am I right?
| Perhaps it could be mentioned that if we consider the probability $P_{n}$ that two randomly chosen integers in $\{1, 2, \dots, n \}$ are coprime
$$
P_{n} = \frac{\lvert \{(a, b) : 1 \le a, b \le n, \gcd(a,b) =1 \}\rvert}{n^{2}},
$$
then
$$
\lim_{n \to \infty} P_{n} = \frac{6}{\pi^{2}}.
$$
See this Wikipedia article.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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Why are $\log$ and $\ln$ being used interchangeably? A definition for complex logarithm that I am looking at in a book is as follows -
$\log z = \ln r + i(\theta + 2n\pi)$
Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the log will make a difference to the answer.
It also says a few lines late... | "$\log$" with no base generally means base the base is $e$, when the topic is mathematics, just as "$\exp$" with no base means the base is $e$. In computer programming languages, "$\log$" also generally means base-$e$ log.
On calculators, "$\log$" with no base means the base is $10$ because calculators are designed by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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General solution using Euclidean Algorithm
I was able to come up with the integer solution that they also have in the textbook using the same method they used but I am really puzzled how they come up with a solution for all the possible integer combinations...how do they come up with that notation/equation that repre... | A general rule in life: When you have a linear equation(s) of the form $f(x_1,x_2,\dots, x_n)=c$, find one solution to the equation and then find a general solution to $f(x_1,\dots,x_n)=0$ and now you can obtain the general solution for the initial equation by adding the special solution you found with the general solu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Pointwise limit of continuous functions not Riemann integrable The following is an exercise from Stein's Real Analysis (ex. 10 Chapter 1). I know it should be easy but I am somewhat confused at this point; it mostly consists of providing the Cantor-like construction for continuous functions on the interval $[0,1]$ whos... | Take a point $c\in C'$ and any open interval $I$ containing $c$.
Then there is an open interval $D\subseteq I $ that was removed in the construction of $C'$.
Indeed, since $C'$ has no isolated points, there is a point $y\in C'\cap I$ distinct from $x$. Between $x$ and $y$, there is an open interval removed from the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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group of order 28 is not simple I have a proof from notes but I don't quite understand the bold part:
Abelian case: $a \in G / \{1\}$. If $\langle a\rangle \neq G$, then we are done. If $\langle a\rangle = G$, then $\langle a^4\rangle $ is a proper normal subgroup of $G$. General case: WLOG we can assume $G \neq Z(G)$.... | Reading the class equation modulo 7 gives the existence of one $x$ such that $\frac{|G|}{|C_G(x)|}$ is NOT divisible by 7. Hence 7 divides $|C_G(x)|$. Now the factors of the numerator $|G|$ are 1, 2, 4, 7 $\cdots$. Since $\frac{|G|}{|C_G(x)|}$ cannot be 1 and cannot divide 7, the only possibilities are 2 and 4.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/108789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Number of 5 letter words over a 4 letter group using each letter at least once Given the set $\{a,b,c,d\}$ how many 5 letter words can be formed such that each letter is used at least once?
I tried solving this using inclusion - exclusion but got a ridiculous result:
$4^5 - \binom{4}{1}\cdot 3^5 + \binom{4}{2}\cdot 2^5... | Your mistake is in the arithmetic. What you think comes out to 2341 really does come out to 240.
$4^5=1024$, $3^5=243$, $2^5=32$, $1024-(4)(243)+(6)(32)-4=1024-972+192-4=1216-976=240$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Upper bounds on the size of $\operatorname{Aut}(G)$ Any automorphism of a group $G$ is a bijection that fixes the identity, so an easy upper bound for the size of $\operatorname{Aut}(G)$ for a finite group $G$ is given by
\begin{align*}\lvert\operatorname{Aut}(G)\rvert \leq (|G| - 1)! \end{align*}
This inequality is a... | I believe this is an exercise in Wielandt's permutation groups book.
$\newcommand{\Aut}{\operatorname{Aut}}\newcommand{\Sym}{\operatorname{Sym}}\Aut(G) \leq \Sym(G\setminus\{1\})$ and so if $|\Aut(G)|=(|G|-1)!$, then $\Aut(G) = \Sym(G\setminus\{1\})$ acts $|G|-1$-transitively on the non-identity elements of G. This me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 6,
"answer_id": 5
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Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$ $$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$
Does it converge or diverge?
Can we have a rigorous proof that is not probabilistic?
For reference, this question is supposedly a mix of real analysis and calculus.
| The values for which $\sin(n)$ is close to $1$ (say in an interval $[1-\varepsilon ; 1]$) are somewhat regular :
$1 - \varepsilon \le \sin(n)$ implies that there exists an integer $k(n)$ such that
$n = 2k(n) \pi + \frac \pi 2 + a(n)$ where $|a(n)| \leq \arccos(1- \varepsilon)$.
As $\varepsilon \to 0$, $\arccos(1- \vare... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 5,
"answer_id": 1
} |
Integration analog of automatic differentiation I was recently looking at automatic differentiation.
*
*Does something like automatic differentiation exist for integration?
*Would the integral be equivalent to something like Euler's method? (or am I thinking about it wrong?)
edit:
I am looking at some inherited ... | If I'm reading your question correctly: I don't believe there is an algorithm that, given the algorithm for your function to be integrated and appropriate initial conditions, will give an algorithm that corresponds to the integral of your original function.
However: you might wish to look into the Chebfun project by Tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Graph-Minor Theorem for Directed Graphs? Suppose that $\vec{G}$ is a directed graph and that $G$ is the undirected graph obtained from $\vec{G}$ by forgetting the direction on each edge. Define $\vec{H}$ to be a minor of $\vec{G}$ if $H$ is a minor of $G$ as undirected graphs and direction on the edges of $\vec{H}$ are... | I think the answer is yes, see 10.5 in Neil Robertson and Paul D. Seymour. Graph minors. xx. wagner’s conjecture. Journal of Combinatorial Theory, 92:325–357, 2004. and the preceding section:
As a corollary, we deduce the following form of Wagner’s conjecture for directed graphs (which immediately implies the standard... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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What is a Gauss sign? I am reading the paper
"A Method for Extraction of Bronchus Regions from 3D Chest X-ray
CT Images by Analyzing Structural Features of the Bronchus" by Takayuki KITASAKA, Kensaku MORI, Jun-ichi HASEGAWA and Jun-ichiro TORIWAKI
and I run into a term I do not understand:
In equation (2), when we ... | From the context (a change of scale using discrete units), this should certainly mean floor as on page 5 of Gauss's Werke 2
per signum $[x]$ exprimemus integrum ipsa $x$ proxime minorem, ita ut $x-[x]$ semper fiat quantitas positiva intra limites $0$ et $1$ sita
i.e. the next lower integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/109179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Positive semi-definite matrix Suppose a square symmetric matrix $V$ is given
$V=\left(\begin{array}{ccccc}
\sum w_{1s} & & & & \\
& \ddots & & -w_{ij} \\
& & \ddots & & \\
& -w_{ij} & & \ddots & \\
& & & & \sum w_{ns}
\end{array}\right) \in\mathbb{R}^{n\times n},$
with values $w_{ij}> 0$... | Claim: For a symmetric real matrix $A$, then $tr(X^TAX)\ge 0$ for all $X$ if and only if $A$ is positive semidefinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/109231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The set of limit points of an unbounded set of ordinals is closed unbounded. Let $\kappa$ be a regular, uncountable cardinal. Let $A$ be an unbounded set, i.e. $\operatorname{sup}A=\kappa$. Let $C$ denote the set of limit points $< \kappa$ of $A$, i.e. the non-zero limit ordinals $\alpha < \kappa$ such that $\operatorn... | Fix $\xi\in \kappa$, since $A$ is unbounded there is a $\alpha_0\in A$ so that $\xi<\alpha_0$. Now, construct recursively a strictly increasing sequence $\langle \alpha_n: n\in \omega\rangle$. Let $\alpha=\sup\{\alpha_n: n\in \omega\}.$ Since $\kappa$ is regular and uncountable, we have $\alpha<\kappa.$ It is also easy... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving modular equations Is there a procedure to solve this or is it strictly by trial and error?
$5^x \equiv 5^y \pmod {39}$ where $y > x$.
Thanks.
| Hint: since $39 = 3\cdot 13$ we can compute the order of $5\ ({\rm mod}\ 39)$ from its order mod $3$ and mod $13$.
First, mod $13\!:\ 5^2\equiv -1\ \Rightarrow\ 5^4\equiv 1;\ \ $ Second, mod $3\!:\ 5\equiv -1\ \Rightarrow\ 5^2\equiv 1\ \Rightarrow\ 5^4\equiv 1 $.
Thus $\:3,13\ |\ 5^4-1\ \Rightarrow\ {\rm lcm}(3,13)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ A logarithm of base b for x is defined as the number u such that $b^u=x$. Thus, the logarithm with base $e$ gives us a $u$ such that $e^u=b$.
In the presentations that I have come across, the author starts with the fundamental property $f(xy) = f(x)+f(y)$ and goes on to... | The following properties uniquely determine the natural log:
1) $f(1) = 0$.
2) $f$ is continuous and differentiable on $(0, \infty)$ with $f'(x) = \frac{1}{x}$.
3) $f(xy) = f(x) + f(y)$
We will show that the function $f(x) = \int_1^x \frac{1}{t} dt$ obeys properties
1,2, and 3, and is thus the natural log.
1) This is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate this limit? Doesn't seem to be difficult, but still can't get it.
$\displaystyle\lim_{x\rightarrow 0}\frac{10^{-x}-1}{x}=\ln10$
| It's worth noticing that we can define the natural logarithm as
$$\log x = \lim_{h \to 0} \frac{x^h-1}{h}$$
So in your case you have
$$ \lim_{h \to 0} \frac{10^{-h}-1}{h}= \lim_{h \to 0} \frac{\frac{1}{10}^{h}-1}{h}=-\log10$$
This result holds because we have that
$$ \lim_{h \to 0} \frac{x^h-1}{h} =\frac{0}{0}$$
So we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Rate of convergence of a sequence in $\mathbb{R}$ and Big O notation From Wikipedia
$f(x) = O(g(x))$ if and only if there exists a positive real number
$M$ and a real number $x_0$ such that $|f(x)| \le \; M |g(x)|\mbox{
for all }x>x_0$.
Also from Wikipedia
Suppose that the sequence $\{x_k\}$ converges to the num... | To answer your added question,
from the definition,
$x_n$ converges to $L$ if and only if
$|x_n-L| \to 0$ as $n \to \infty$.
The existence of a positive c such that
$c < 1$ and $|x_{n+1}-L| \le c|x_n-L|$
is sufficient for convergence, but not necessary.
For example, if $x_n = 1/(\ln n)$,
then $x_n \to 0$, but there is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$ I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one:
In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the
coefficients of $x^3$?
| You can just multiply it out. Alternatively, you can reason the terms are of the form $1^a(\frac x2)^b(\frac 1{\sqrt x})^c$ with $a+b+c=8, b-\frac c2=3$. Then $c=2b-6$, so $a+3b=14$ and $a$ needs to be $2 \text{ or } 5$. Then you need the multinomial coefficient as stated by Suresh.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/109748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
How can we find the values that a (divergent!) series tends to? Suppose we are given a series that diverges. That's right, diverges. We may interest ourselves in the limiting function(s) of its behavior.
For instance, given the power series:$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots$$
I am interested in finding the ... | Note that the usual definition of the infinite sum is a very different kind of thing from an ordinary finite sum. It introduces ideas from topology, analysis or metric spaces which aren't present in the original definition of sum. So when generalising from finite sums to infinite sums we have quite a bit of choice in h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$ Given that n is a positive integer show that $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$.
I'm thinking that I should be using the property of gcd that says if a and b are integers then gcd(a,b) = gcd(a+cb,b). So I can do things like decide that $\gcd(n^3 + 1, n^2 + 2) = ... | Let $\:\rm d = (n^3+1,\:n^2+2).\:$ Observe that $\rm \ d \in \{1,\:3,\:9\} \iff\ d\:|\:9\iff 9\equiv 0\pmod d\:.$
mod $\rm (n^3\!-a,n^2\!-b)\!:\ a^2 \equiv n^6 \equiv b^3\:$ so $\rm\:a=-1,\:b = -2\:\Rightarrow 1\equiv -8\:\Rightarrow\: 9\equiv 0\:. \ \ $ QED
Or, if you don't know congruence arithmetic, since $\rm\: x-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the supremum of an ordinal the next ordinal? I apologize for this naive question.
Let $\eta$ be an ordinal. Isn't the supremum of $\eta$ just $\eta+1$? If this is true, the supremum is only necessary if you conisder sets of ordinals.
| There are two notions of supremum in sets ordinals, let $A$ be a set of ordinals:
*
*$\sup^+(A)=\sup\{\alpha+1\mid\alpha\in A\}$,
*$\sup(A) =\sup\{\alpha\mid\alpha\in A\}$.
Since most of the time we care about supremum below limit ordinals (eg. $A=\omega$) the notions coincide. If $A=\{\alpha\}$ then indeed $\sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sum of alternating reciprocals of logarithm of 2,3,4... How to determine convergence/divergence of this sum?
$$\sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)}$$
Why cant we conclude that the sum $\sum_{k=2}^\infty (-1)^k\frac{k}{p_k}$, with $p_k$ the $k$-th prime, converges, since $p_k \sim k \cdot \ln(k)$ ?
| You are correct.
The alternating series test suffices, no need to look at the dirichlet test.
Think about it, the sequence $|a|\rightarrow 0$ at $n\rightarrow\infty$, and decreases monotonically ($a_n>a_{n+1}$). That means that you add and remove terms that shrink to $0$. What you add you remove partially in the next t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Inverse function that takes connected set to non-connected set I've been struggling with providing examples of the following:
1) A continuous function $f$ and a connected set $E$ such that $f^{-1}(E)$ is not connected
2) A continuous function $g$ and a compact set $K$ such that $f^{-1}(K)$ is not compact
| Take any space $X$ which is not connected and not compact. For example, you could think of $\mathbf R - \{0\}$. Map this to a topological space consisting of one point. [What properties does such a space have?]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/110069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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CLT for arithmetic mean of centred exp.distributed RV $X_1, X_2,\ldots$ are independent, exponentially distributed r. variables with $m_k:=EX_k=\sqrt{2k}$, $v_k:=\operatorname{Var} X_k=2k$.
I want to analyse the weak convergence of $Y_n=\dfrac{\sum_{i=1}^n (X_i−m_i)}{n}$.
CLT: When the Lindeberg Condition
$b^{−2}_n\su... | Let $x_{k,n}=\mathrm E((X_k-m_k)^2:|X_k-m_k|\geqslant\varepsilon b_n)$. Since $X_k/m_k$ follows the distribution of a standard exponential random variable $X$,
$$
x_{k,n}=m_k^2\mathrm E((X-1)^2:(X-1)^2\geqslant\varepsilon^2 b_n^2/m_k^2).
$$
In particular, $x_{k,n}\leqslant m_k^2x_{n,n}$ for every $k\leqslant n$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Irrationality of "primes coded in binary" For fun, I have been considering the number
$$
\ell := \sum_{p} \frac{1}{2^p}
$$
It is clear that the sum converges and hence $\ell$ is finite. $\ell$ also has the binary expansion
$$
\ell = 0.01101010001\dots_2
$$
with a $1$ in the $p^{th}$ place and zeroes elsewhere. I ... | That $\ell$ is irrational is clear. There are arbitrarily large gaps between consecutive primes, so the binary expansion of $\ell$ cannot be periodic. Any rational has a periodic binary expansion.
The fact that there are arbitrarily large gaps between consecutive primes comes from observing that if $n>1$, then all of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
Derivation of asymptotic solution of $\tan(x) = x$. An equation that seems to come up everywhere is the transcendental $\tan(x) = x$. Normally when it comes up you content yourself with a numerical solution usually using Newton's method. However, browsing today I found an asymptotic formula for the positive roots $x$:
... | You may be interested in N. G. de Bruijn's book Asymptotic Methods in Analysis, which treats the equation $\cot x = x$. What follows is essentially a minor modification of that section in the book.
The central tool we will use is the Lagrange inversion formula. The formula given in de Bruijn differs slightly from the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 1,
"answer_id": 0
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Do the algebraic and geometric multiplicities determine the minimal polynomial? Let $T$ denote some linear transformation of a finite-dimensional space $V$ (say, over $\mathbb{C}$).
Suppose we know the eigenvalues $\{\lambda_i\}_i$ and their associated algebraic multiplicities $\{d_i\}_i$ and geometric multiplicities $... | No, the algebraic and geometric multiplicities do not determine the minimal polynomial. Here is a counterexample: Consider the Jordan matrices $J_1, J_2$:
$$J_1 = \left(
\begin{array}{cccc}
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1
\end{array}
\right)
~~
J_2 =
\left(
\begin{array}{cccc}
1 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
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Number of Solutions of $3\cos^2(x)+\cos(x)-2=0$ I'm trying to figure out how many solutions there are for
$$3\cos^2(x)+\cos(x)-2=0.$$
I can come up with at least two solutions I believe are correct, but I'm not sure if there is a third.
| Unfortunately, there are infinitely many solutions. Note, for example, that $\pi$ is a solution. Then we also have that $\pi + 2k \pi$ is a solution for all $k$. But between $0$ and $2\pi$, there are 3 solutions.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
$$\displaystyle \int\limits_0^\infty {\frac{{{x^1}}}{{1 + {x^3}... | The general formula (for $m > n+1$ and $n \ge 0$) is $\frac{\pi}{m} \csc\left(\frac{\pi (n+1)}{m}\right)$. IIRC the usual method involves a wedge-shaped contour of angle $2 \pi/m$.
EDIT: Consider $\oint_\Gamma f(z)\ dz$ where $f(z) = \frac{z^n}{1+z^m}$ (using the principal branch if $m$ or $n$ is a non-integer) and $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "109",
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Calculating the percentage difference of two numbers The basic problem is this: "I have this number x and I ask you to give me another number y. If the number you give me is some percentage c different than my number then I do not want it." Given that you will know x and c, how do you calculate whether or not I should ... | What you're missing is what you want. The difference between your two numbers is clearly $|x-y|$, but the "percentage" depends on how you want to write $|x-y|/denominator$. You could choose for a denominator $|x|$, $|x+y|$, $\max \{x,y\}$, $\sqrt{x^2 + y^2}$, for all I care, it's just a question of choice. Personally I... | {
"language": "en",
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"source": "stackexchange",
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Combinatorics-N boys and M girls are learning acting skills from a theatre in Mumbai. N boys and M girls are learning acting skills from a theatre in Mumbai. To perform a play on ‘Ramayana’ they need to form a group of P actors containing not less than 4 boys and not less than 1 girl. The theatre requires you to write ... | Assume that $M \ge 1$, $N\ge 4$, and $P\ge 5$. In how many ways can we choose $P$ people, with no sex restrictions? We are choosing $P$ people from $M+N$. The number of choices is
$$\tbinom{M+N}{P}.$$
However, choosing $0$, $1$, $2$, or $3$ boys is forbidden.
In how many ways can we choose $0$ boys, and therefore $P... | {
"language": "en",
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Evaluating $\int_0^\infty e^{-x^n}\,\mathrm{d}x$ Is there a general approach to evaluating definite integrals of the form $\int_0^\infty e^{-x^n}\,\mathrm{d}x$ for arbitrary $n\in\mathbb{Z}$? I imagine these lack analytic solutions, so some sort of approximation is presumably required. Any pointers are welcome.
| For $n=0$ the integral is divergent and if $n<0$ $\lim_{n\to\infty}e^{-x^n}=1$ so the integral is not convergent.
For $n>0$ we make the substitution $t:=x^n$, then $$I_n:=\int_0^{+\infty}e^{-x^n}dx=\int_0^{+\infty}e^{—t}t^{\frac 1n-1}\frac 1ndt=\frac 1n\Gamma\left(\frac 1n\right),$$
where $\Gamma(\cdot)$ is the usual ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple algebra. How can you factor out a common factor if it is $\frac{1}{\text{factor}}$ in one of the cases? I'm sure this is simple. I want to pull out a factor as follows...
I have the expression
$$\frac{a(\sqrt{x}) - (b + c)(\frac{1}{\sqrt{x}})c}{x}.$$
It would be useful for me to pull out the $\sqrt{x}$ from the... | $$ \frac{a(\sqrt{x}) - (b + c)({\frac{\sqrt{x}}{x}})c}{x}\;\tag{1}$$
$$=\frac{a(\color{red}{\sqrt{x}}) - (b + c)({\frac{\color{red}{\sqrt{x}}}{x}})c}{\color{red}{\sqrt x}\sqrt x}\;\tag{2}$$
$$=\frac{(\color{red}{\sqrt x})[a - (b + c)({\frac{1}{x}})c]}{\color{red}{\sqrt x}\sqrt x}\;\tag{3}$$
$$=\frac{[a - (b + c)({\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Help with a short paper - cumulative binomial probability estimates I was hoping someone could help me with a brief statement I can't understand in a book.
The problem I have is with the final line of the following section of Lemma 2.2 (on the second page):
Since $|\mathcal{T}_j|$ is bin. distributed with expectation ... | Wikipedia is your friend. In general, when a paper mentions using technique X, if you are not aware of technique X, then look it up. It will be impossible to fill the gap without knowing about X.
In the case at hand, X is the Chernoff bound (also Hoeffding's inequality, and even more names). It's indeed pretty standard... | {
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A contest problem about multiple roots if a is real, what is the only real number that could be a mutiple root of $x^3 +ax+1$=0
No one in my class know how to do it, so i have to ask it here.
| Let the multiple root be $r$, and let the other root be $s$. If $r$ is to be real, then $s$ must be real also. From Vieta's formulas, we have $2r + s = 0$ and $r^2s = -1$. The first equation gives $s = -2r$, which we plug into the second equation to get $r^2s = -2r^3 = -1$, so $r = \boxed{\left(\frac12\right)^{1/3}}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combining Taylor expansions How do you taylor expand the function $F(x)={x\over \ln(x+1)}$ using standard results? (I know that WA offers the answer, but I want to know how to get it myself.) I know that $\ln(x+1)=x-{x^2\over 2}+{x^3\over 3}+…$ But I don't know how to take the reciprocal. In general, given a function $... | You have
$$F(x)=\frac{x}{\sum_{n\ge 1}\frac{(-1)^{n+1}}nx^n}=\frac1{\sum_{n\ge 0}{\frac{(-1)^n}{n+1}}x^n}=\frac1{1-\frac{x}2+\frac{x^2}3-+\dots}\;.$$
Suppose that $F(x)=\sum\limits_{n\ge 0}a_nx^n$; then you want
$$1=\left(1-\frac{x}2+\frac{x^2}3-+\dots\right)\left(a_0+a_1x+a_2x^2+\dots\right)\;.$$
Multiply out and equ... | {
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Area of a trapezoid from given the two bases and diagonals Find the area of trapezoid with bases $7$ cm and $20$ cm and diagonals $13$ cm and $5\sqrt{10} $ cm.
My approach:
Assuming that the bases of the trapezoid are the parallel sides, the solution I can think of is a bit ugly,
*
*Find the other two non-parallel ... | Let's denote $a=20$ , $b=7$ ,$d_1=13$ , $d_2=5 \sqrt{10}$ , (see picture below)
You should solve following system of equations :
$\begin{cases}
d_1^2-(b+x)^2=d_2^2-(b+y)^2 \
a-b=x+y
\end{cases}$
After you find values of $x$ and $y$ calculate $h$ from one of the following equations :
$h^2=d_2^2-(b+y)^2$ , or
$h^2= d_... | {
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How to get from $a\sqrt{1 + \frac{b^2}{a^2}}$ to $\sqrt{a^2 + b^2}$ I have the following expression: $a\sqrt{1 + \frac{b^2}{a^2}}$. If I plug this into Wolfram Alpha, it tells me that, if $a, b$ are positive, this equals $\sqrt{a^2 + b^2}$.
How do I get that result? I can't see how that could be done. Thanks
| $$a\sqrt{1 + \frac{b^2}{a^2}}$$
$$=a\sqrt{\frac{a^2 + b^2}{a^2}}$$
$$=a\frac{\sqrt{a^2 + b^2}}{|a|}$$
So when $a$ and $b$ are positive, $|a|=a$. Hence:
$$=\sqrt{a^2 + b^2}$$
Without the assumption:
$$\sqrt{a^2} =|a|=\begin{cases} a && a \geq 0\\ -a &&a < 0\\ \end{cases}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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How to deal with multiplication inside of integral? I have an undefined integral like this:
\begin{aligned}
\ \int x^3 \cdot \sin(4+9x^4)dx
\end{aligned}
I have to integrate it and I have no idea where to start. I have basic formulas for integrating but I need to split this equation into two or to do something else.
| Note that $$(4+9x^4)' = 36x^3$$
So that your integral becomes
$$\int x^3 \sin(4+9x^4)dx$$
$$\dfrac{1}{36}\int 36x^3 \sin(4+9x^4)dx$$
$$\dfrac{1}{36}\int \sin u du$$
Which you can easily solve.
| {
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"source": "stackexchange",
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Prove by induction that $n!>2^n$
Possible Duplicate:
Proof the inequality $n! \geq 2^n$ by induction
Prove by induction that $n!>2^n$ for all integers $n\ge4$.
I know that I have to start from the basic step, which is to confirm the above for $n=4$, being $4!>2^4$, which equals to $24>16$.
How do I continue though.... | Hint: prove inductively that a product is $> 1$ if each factor is $>1$. Apply that to the product $$\frac{n!}{2^n}\: =\: \frac{4!}{2^4} \frac{5}2 \frac{6}2 \frac{7}2\: \cdots\:\frac{n}2$$
This is a prototypical example of a proof employing multiplicative telescopy. Notice how much simpler the proof becomes after trans... | {
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"timestamp": "2023-03-29T00:00:00",
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Linear Algebra: Find a matrix A such that T(x) is Ax for each x I am having difficulty solve this problem in my homework:
(In my notation, $[x;y]$ represents a matrix of 2 rows, 1 column)
Let $\mathbf{x}=[x_1;x_2]$, $v_1$=[−3;5] and $v_2=[7;−2]$ and let $T\colon\mathbb{R}^2\to\mathbb{R}^2$ be a linear transformation ... | If I understand you correctly, I would say that
$$A = \left(\begin{array}{rr}-3&7\\5&-2\end{array}\right) \ \textrm{and} \ x'=Ax.$$
You can see this if you use
$$x' = \left(\begin{array}{cc}x_1\\x_2\end{array}\right).$$
Then $$x_1'= -3\cdot x_1 + 7\cdot x_2 = x_1 \cdot v_{11} + x_2\cdot v_{21}$$ and $$x_2'= 5\cdot x_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What does $\sin^{2k}\theta+\cos^{2k}\theta=$?
What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to?
Besides Mathematical Induction,more solutions are desired.
| If you let $z_k=\cos^k(\theta)+i\sin^k(\theta)\in\Bbb C$, it is clear that
$$
\cos^{2k}(\theta)+\sin^{2k}(\theta)=||z_k||^2.
$$
When $k=1$ the complex point $z_1$ describes (under the usual Argand-Gauss identification $\Bbb C=\Bbb R^2$) the circumference of radius $1$ centered in the origin, and your expression gives... | {
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"source": "stackexchange",
"question_score": "3",
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Density function with absolute value Let $X$ be a random variable distributed with the following density function:
$$f(x)=\frac{1}{2} \exp(-|x-\theta|) \>.$$
Calculate: $$F(t)=\mathbb P[X\leq t], \mathbb E[X] , \mathrm{Var}[X]$$
I have problems calculating $F(t)$ because of the absolute value. I'm doing it by case stat... | The very best thing you can do in solving problems such as these is to sketch the given density function first. It does not have to be a very accurate sketch: if you drew a peak of $\frac{1}{2}$ at $x=\theta$ and decaying curves on either side, that's good enough!
Finding $F_X(t)$:
*
*Pick a number $t$ that is sma... | {
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Solution to Locomotive Problem (Mosteller, Fifty Challenging Problems in Probability) My question concerns the solution Professor Mosteller gives for the Locomotive Problem in his book, Fifty Challenging Problems in Probability. The problem is as follows:
A railroad numbers its locomotives in order 1, 2, ..., N. One da... | Choosing $2\times 60 - 1$ gives an unbiased estimate of $N$.
Choosing $60$ gives a maximum likelihood estimate of $N$.
But these two types of estimator are often different, and indeed this example is the one used by Wikipedia to show that the bias of maximum-likelihood estimators can be substantial.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Lemma vs. Theorem I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a little about the ethymology but mainly about why we choose Lemma for some findings, and Theorem for oth... | There is no mystery regarding the use of these terms: an author of a piece of mathematics will label auxiliary results that are accumulated in the service of proving a major result lemmas, and will label the major results as propositions or (for the most major results) theorems. (Sometimes people will not use the inte... | {
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"source": "stackexchange",
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Examples of patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of... | This might be a simple example.
If we inscribe a circle of radius 1 in a square of side 2, the ratio of the area of the circle to the square is $\frac{\pi}{4}$. You can show that any time we put a square number of circles into this square, the ratio of the area of the circles to that of the square is (for the simple sy... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "612",
"answer_count": 41,
"answer_id": 24
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Solving the equation $- y^2 - x^2 - xy = 0$ Ok, this is really easy and it's getting me crazy because I forgot nearly everything I knew about maths!
I've been trying to solve this equation and I can't seem to find a way out.
I need to find out when the following equation is valid:
$$\frac{1}{x} - \frac{1}{y} = \frac{1}... | $x^2-xy+y^2=(x+jy)(x+j^2y)$ so $x=y(1+\sqrt{-3})/2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/111486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
Suppose $f$ is a Riemann integrable function on $[0,1]$. Prove that $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
This is what I am thinking: Fix $n$. Then by Jensen's Inequality we have $$0\leq\left(\int_0^1x^nf(x)dx\right)^2 \leq \left(\int_0^1x^{... | That looks great. If someone doesn't know Jensen's inequality, this is still seen just with Cauchy-Schwarz. Another quick method is the dominated convergence theorem. Gerry's and Peters answers are both far simpler though.
| {
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"source": "stackexchange",
"question_score": "5",
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Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right? Is $3^x \lt 1 + 2^x + 3^x \lt 3 \cdot 3^x$ right?
This is from my lecture notes which is used to solve:
But when $x = 0$, $(1 + 2^x + 3^x = 3) \gt (3^0 = 1)$? The thing is how do I choose which what expression should go on the left & right side?
| When $x=0$, the left side $3^0=1$, the center is $3$ as you say, and the right side is $3\cdot 3^0=3 \cdot 1=3$ so the center and right sides are equal. But you want this for large $x$, so could restrict the range to $x \gt 1$, say.
| {
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"timestamp": "2023-03-29T00:00:00",
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Notation of the summation of a set of numbers Given a set of numbers $S=\{x_1,\dotsc,x_{|S|}\}$, where $|S|$ is the size of the set, what would be the appropriate notation for the sum of this set of numbers? Is it
$$\sum_{x_i \in S} x_i
\qquad\text{or}\qquad
\sum_{i=1}^{|S|} x_i$$ or something else?
| Say I had a set A, under an operation with the properties of $+$, then $$\sum_{i\in A} x_i$$ is how I write it.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $A,B\in M(2,\mathbb{F})$ and $AB=I$, then $BA=I$ This is Exercise 7, page 21, from Hoffman and Kunze's book.
Let $A$ and $B$ be $2\times 2$ matrices such that $AB=I$. Prove that
$BA=I.$
I wrote $BA=C$ and I tried to prove that $C=I$, but I got stuck on that. I am supposed to use only elementary matrices to solve... | $AB= I$, $Det(AB) = Det (A) . Det(B) = 1$. Hence $Det(B)\neq 0$ Hence $B$ is invertible.
Now let $BA= C$ then we have $BAB= CB$ which gives $B= CB$ that is $B. B^{-1} = C$ this gives $ C= I$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
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Fibonacci numbers modulo $p$ If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.
Who first proved this? Is there a proof simple enough for an undergraduate number theory course? (We will get to quadratic... | Here's a proof that only uses a little Galois theory of finite fields (and QR). I don't know if it's any of the proofs referenced by Gerry. Recall that
$$F_n = \frac{\phi^n - \varphi^n}{\phi - \varphi}$$
where $\phi, \varphi$ are the two roots of $x^2 = x + 1$. Crucially, this formula remains valid over $\mathbb{F}_{p^... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Resources for learning Elliptic Integrals During a quiz my Calc 3 professor made a typo. He corrected it in class, but he offered a challenge to anyone who could solve the integral.
The (original) question was:
Find the length of the curve as described by the vector valued function $\vec{r} = \frac{1}{3}t^{3}\vec{i} +... | There are plenty of places to look (for example, most any older 2-semester advanced undergraduate "mathematics for physicists" or "mathematics for engineers" text), but given that you're in Calculus III, some of these might be too advanced. If you can find a copy (your college library may have a copy, or might be able ... | {
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"timestamp": "2023-03-29T00:00:00",
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Why are perpendicular bisectors 'lines'? Given two points $p$ and $q$ their bisector is defined to be $l(p,q)=\{z:d(p,z)=d(q,z)\}$.
Due to the construction in Euclidean geometry, we know that $l(p,q)$ is a line, that is, for $x,y,z\in l(p,q)$, we have $d(x,y)+d(y,z)=d(x,z)$, which charactorizes lines.
I wonder whether... | Let $A$ and $B$ be the two given points and let $M$ be the midpoint of $AB$, i.e., $M\in A\vee B$ and $d(M,A)=d(M,B)$. Let $X\ne M$ be an arbitrary point with $d(X,A)=d(X,B)$. Then the triangles $\Delta(X,A,M)$ and $\Delta(X,B,M)$ are congruent as corresponding sides have equal length. It follows that $\angle(XMA)=\ang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can there exist a non-constant continuous function that has a derivative of zero everywhere? Somebody told me that there exists a continuous function with a derivative of zero everywhere that is not constant.
I cannot imagine how that is possible and I am starting to doubt whether it's actually true. If it is true, cou... | Since there are no restrictions on the domain, it is actually possible. Let $f:(0,1)\cup(2,3)\to \mathbb R$ be defined by $f(x)=\left\{
\begin{array}{ll}
0 & \mbox{if } x \in (0,1) \\
1 & \mbox{if } x\in (2,3)
\end{array}
\right.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Simplest Example of a Poset that is not a Lattice A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a partially ordered set can fail to be a lattice. In $\mathbb{R}$, for e... | The set $\{x,y\}$ in which $x$ and $y$ are incomparable is a poset that is not a lattice, since $x$ and $y$ have neither a common lower nor common upper bound. (In fact, this is the simplest such example.)
If you want a slightly less silly example, take the collection $\{\emptyset, \{0\}, \{1\}\}$ ordered by inclusion.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 6,
"answer_id": 1
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$C[0,1]$ is not Hilbert space Prove that the space $C[0,1]$ of continuous functions from $[0,1]$ to $\mathbb{R}$ with the inner product $ \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad $ is not Hilbert space.
I know that I have to find a Cauchy sequence $(f_n)_n$ which converges to a function $f$ which is not con... | You are right to claim that in order to prove that the subspace $C[0,1]$ of $L^2[0,1]$ is not complete, it is sufficient to "find [in $C[0,1]$] a Cauchy sequence $(f_n)_n$ [i.e. Cauchy for the $L^2$-norm] which converges [in $L^2[0,1]$] to a function $f$ which is not continuous". It will even be useless to check that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 2
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Solving the system $\sum \sin = \sum \cos = 0$. Can we solve the system of equations:
$$\sin \alpha + \sin \beta + \sin \gamma = 0$$
$$\cos \alpha + \cos \beta + \cos \gamma = 0$$
?
(i.e. find the possible values of $\alpha, \beta, \gamma$)
| Developing on Gerenuk's answer, you could consider the complex numbers
$$ z_1=\cos \alpha+i\sin \alpha,\ z_2=\cos \beta+i\sin\beta,\ z_3=\cos \gamma+i\sin \gamma$$
Then you know that $z_1,z_2,z_3$ are on the unit circle, and the centroid of the triangle formed by the points of afixes $z_i$ is of afix $\frac{z_1+z_2+z_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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Is the domain of an one-to-one function a set if the target is a set? This is probably very naive but suppose I have an injective map from a class into a set, may I conclude that the domain of the map is a set as well?
| If a function $f:A\to B$ is injective one, we can assume without loss of generality that $f$ is surjective too (by passing to a subclass of $B$), therefore $f^{-1}:B\to A$ is also a bijection.
If $B$ is a set then every subclass of $B$ is a set, so $f^{-1}:B\to A$ is a bijection from a set, and by the axiom of replacem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Basis for adjoint representation of $sl(2,F)$ Consider the lie algebra $sl(2,F)$ with standard basis $x=\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$, $j=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $h=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$.
I want to find the casimir element of the adjoint representation o... | A representation for a Lie algebra $\mathfrak{g}$ is a Lie algebra homomorphism $\varphi:\mathfrak{g} \to \mathfrak{gl}(V)$ for some vector space $V$.
Of course, every representation corresponds to a module action. In the case of this representation the module action would be $g \cdot v = \varphi(g)(v)$.
It is not clea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are the possible values for $\gcd(a^2, b)$ if $\gcd(a, b) = 3$? I was looking back at my notes on number theory and I came across this question.
Let $a$, $b$ be positive integers such that $\gcd(a, b) = 3$. What are the possible values for $\gcd(a^2, b)$?
I know it has to do with their prime factorization decompos... | If $p$ is a prime, and $p|a^2$, then $p|a$; thus, if $p|a^2$ and $p|b$, then $p|a$ and $p|b$, hence $p|\gcd(a,b) = 3$. So $\gcd(a^2,b)$ must be a power of $3$.
Also, $3|a^2$ and $3|b$, so $3|\gcd(a^2,b)$; so $\gcd(a^2,b)$ is a multiple of $3$.
If $3^{2k}|a^2$, then $3^k|a$ (you can use prime factorization here); so if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The inclusion $j:L^{\infty}(0,1)\to L^1(0,1)$ is continuous but not compact. I'm stuck on this problem, namely I cannot find a bounded subset in $L^\infty(0,1)$ such that it is not mapped by the canonical inclusion $$j: L^\infty(0,1)\to L^1(0,1)$$ onto a relatively compact subset in $L^1(0,1)$. Can anybody provide me a... | This is actually just a variant of a special case of NKS’s example, but it may be especially easy to visualize with this description.
For $n\in\mathbb{Z}^+$ and $x\in(0,1)$ let $f_n(x)$ be the $n$-th bit in the unique non-terminating binary expansion of $x$. Then $\|f_n\|_\infty=1$, but $\|f_n-f_m\|_1=\frac12$ whenever... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Infinite distinct factorizations into irreducibles for an element Consider the factorization into irreducibles of $6$ in $\mathbb{Z}[\sqrt{-5}]$. We have $6=2 \times 3$ and $6=(1+\sqrt{-5}) \times (1-\sqrt{-5})$, i.e. $2$ distinct factorizations. And,
$$6^2=3 \times 3\times2\times2$$
$$=(1+\sqrt{-5}) \times (1-\sqrt{... | If you are only interested in behaviour in the ring of integers of a number field (such as $\mathbb{Z}[\sqrt{-5}]$) then you will never get infinitely many different factorisations of an element.
These different factorisations come from reordering the (finitely many) prime ideals in the unique factorisation of the idea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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For $x+y=n$, $y^x < x^y$ if $x(updated)
I'd like to use this property for my research, but it's somewhat messy to prove.
$$\text{For all natural number $x,y$ such that $x+y=n$ and $1<x<y<n$, then $y^x < x^y$}.$$
For example, let $x=3, y=7$. Then $y^x = y^3 = 343$ and $x^y = 3^7 = 2187$. Any suggestion on how to prove ... | I proved this in the special case $x = 99, y = 100$, here. As others have pointed out, what you really want to hold is the following:
Statement: Let $x, y \in \mathbb{R}$. Then $y > x > e$ implies $x^y > y^x$.
Proof:. Write $y = x + z$, where $z > 0$. Then,
$$\begin{align}
x^y > y^x &\iff x^x x^z > y^x
\\
&\iff x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
"Every linear mapping on a finite dimensional space is continuous" From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector spaces (not necessarily the same), as along as the domain is fin... | The special case of a linear transformations $A: \mathbb{R}^n \to \mathbb{R}^n$ being continuous leads nicely into the definition and existence of the operator norm of a matrix as proved in these notes.
To summarise that argument, if we identify $M_n(\mathbb{R})$ with $\mathbb{R^{n^2}}$, and suppose that $v \in \mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "59",
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What is the Jacobian? What is the Jacobian of the function $f(u+iv)={u+iv-a\over u+iv-b}$?
I think the Jacobian should be something of the form $\left(\begin{matrix}
{\partial f_1\over\partial u} & {\partial f_1\over\partial v} \\
{\partial f_2\over\partial u} & {\partial f_2\over\partial v}
\end{matrix}\righ... | You could just write $(u+iv−(a_1+a_2 i))/(u+iv−(b_1+b_2 i))$ where $u,v,a_1,a_2,b_1,b_2$ are real. Then multiply the numerator and denominator by the complex conjugate of the denominator to find the real and imaginary parts.
Then later, exploit the Cauchy--Riemann equations to conclude that the matrix must have the fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Indefinite integral of $\cos^{3}(x) \cdot \ln(\sin(x))$ I need help. I have to integrate $\cos^{3} \cdot \ln(\sin(x))$ and I don´t know how to solve it. In our book it is that we have to solve using the substitution method. If somebody knows it, you will help me..please
| Substitute :
$\sin x =t \Rightarrow \cos x dx =dt$ , hence :
$I=\int (1-t^2)\cdot \ln (t) \,dt$
This integral you can solve using integration by parts method .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/113184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of $\sum_{0 \le k \le a} {a \choose k} {b \choose k} = {a+b \choose a}$ $$\sum_{0 \le k \le a}{a \choose k}{b \choose k} = {a+b \choose a}$$
Is there any way to prove it directly?
Using that $\displaystyle{a \choose k}=\frac{a!}{k!(a-k)!}$?
| How about this proof? (Actually an extended version of your identity.)
*
*http://en.wikipedia.org/wiki/Chu-Vandermonde_identity#Algebraic_proof
I don't think it is "direct" enough, though...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/113267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Must a measure on $2^{\mathbb{N}}$ be atomless to be a measure on $[0,1]$? This question comes from section 4.4, page 17, of this paper.
Let $\mu$ be a Borel measure on Cantor space, $2^\mathbb{N}$. The authors say that
If the measure is atomless, via the binary expansion of reals we can view it also as a Borel measur... | The existence of the measure on $[0,1]$ has nothing to do with atoms, per se.
Let $\varphi: 2^\mathbb{N}\to [0,1]$ be defined by $\varphi(x)=\sum_{n=0}^\infty {x(n)/2^n}$. This map is Borel measurable, and so for any Borel measure $\mu$ on $2^\mathbb{N}$, the image measure $\mu\circ\varphi^{-1}$ is a Borel measure on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Disprove uniform convergence of $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ in $[0,\infty)$ How would I show that $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ does not uniformly converge in $[0,\infty)$?
I don't know how to approach this problem.
Thank you.
| This is almost the same as Davide' answer:
let
$$f_n(x)={x\over (1+x)^n},\ n\in\Bbb N^+;\ \ \text{ and }\ \ f(x)= \sum\limits_{n=1}^\infty {x\over(1+x)^n}.$$ Since, for $x>0$, the series $\sum\limits_{n=1}^\infty {1\over(1+x)^n}$ is a Geometric series with $r={1\over 1+x}$:
$$
f(x)=x\sum_{n=1}^\infty {1\over(1+x)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
an open ball in $\mathbb{R^n}$ is connected
Show that an open ball in $\mathbb{R^n}$ is a connected set.
Attempt at a Proof: Let $r>0$ and $x_o\in\mathbb{R^n}$. Suppose $B_r(x_o)$ is not connected. Then, there exist $U,V$ open in $\mathbb{R^n}$ that disconnect $B_r(x_o)$. Without loss of generality, let $a\in B_r(x... | $\mathbb{R}=(-\infty,\infty)$, hence it is connected. Since the finite product of connected space is connected, the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/113383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Show that $\tan 3x =\frac{ \sin x + \sin 3x+ \sin 5x }{\cos x + \cos 3x + \cos 5x}$ I was able to prove this but it is too messy and very long. Is there a better way of proving the identity? Thanks.
| More generally, for any arithmetic sequence, denoting $z=\exp(i x)$ and $2\ell=an+2b$, we have
$$\begin{array}{c l}
\blacktriangle & =\frac{\sin(bx)+\sin\big((a+b)x\big)+\cdots+\sin\big((na+b)x\big)}{\cos(bx)+\cos\big((a+b)x\big)+\cdots+\cos\big((na+b)x\big)} \\[2pt]
& \color{Red}{\stackrel{1}=}
\frac{1}{i}\frac{z^b\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Finite Rings whose additive structure is isomorphic to $\mathbb{Z}/(n \mathbb{Z})$ I am having trouble proving the following conjecture: If $R$ is a ring with $1_R$ different from $0_R$ s.t. its additive structure is isomorphic to $\mathbb{Z}/(n \mathbb{Z})$ for some $n$, must $R$ always be isomorphic to the ring $\m... | Combine the following general facts:
For any ring $R$, the prime ring (i.e. the subring generated by $1$) is isomorphic to the quotient of $\mathbb Z$ by the annihilator of $R$ in $\mathbb Z$.
Any cyclic group $R$ is isomorphic to the quotient of $\mathbb Z$ by the annihilator of $R$ in $\mathbb Z$.
(This is Mariano's... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Normal distribution involving $\Phi(z)$ and standard deviation The random variable X has normal distribution with mean $\mu$ and standard deviation $\sigma$. $\mathbb{P}(X>31)=0.2743$ and $\mathbb{P}(X<39)=0.9192$. Find $\mu$ and $\sigma$.
| Hint:
Write,
$$ \tag{1}\textstyle
P[\,X>31\,] =P\bigl[\,Z>{31-\mu\over\sigma}\,\bigr]=.2743\Rightarrow {31-\mu\over\sigma} = z_1
$$
$$\tag{2}\textstyle
P[\,X<39\,] =P\bigl[\,Z<{39-\mu\over\sigma}\,\bigr]=.9192\Rightarrow {39-\mu\over\sigma} =z_2 ,
$$
where $Z$ is the standard normal random variable.
You can fin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a simple expression for this series expansion without a piecewise definition I am doing some practice Calculus questions and I ran into the following problem which ended up having a reduction formula with a neat expansion that I was wondering how to express in terms of a series. Here it is: consider
$$
I_{n} =... | $$
\color{green}{I_n=\sum\limits_{i=2}^{n} (-1)^{n-i}\cdot i\cdot\left(\frac{\pi}{2} \right)^{i-1}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/113655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Why do introductory real analysis courses teach bottom up? A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta\,$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and integration. Only later is the notion of open and closed se... | I'm with Alex Becker, I first learned convergence of sequences, using epsilon and deltas, and only later moved on to continuity of functions. It worked out great for me. I don't believe that the abstraction from topology would be useful at this point. The ideas of "$x$ is near $y$", "choosing $\epsilon$ as small as you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 7,
"answer_id": 6
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Extension and Self Injective Ring Let $R$ be a self injective ring. Then $R^n$ is an injective module. Let $M$ be a submodule of $R^n$ and let $f:M\to R^n$ be an $R$-module homomorphism. By injectivity of $R^n$ we know that we can extend $f$ to $\tilde{f}:R^n\to R^n$.
My question is that if $f$ is injective, can we als... | The question is also true without any commutativity for quasi-Frobenius rings.
Recall that a quasi-Frobenius ring is a ring which is one-sided self injective and one-sided Noetherian. They also happen to be two-sided self-injective and two-sided Artinian.
For every finitely generated projective module $P$ over a quasi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Is the product of symmetric positive semidefinite matrices positive definite? I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices?
My proof of the positive definite case falls... | Actually, one has to be vary careful in the way one interprets the results of Meenakshi and Rajian (referenced in one of the posts above). Symmetry is inherent in their definition of positive definiteness. Thus, their result can be stated very simply as follows: If $A$ and $B$ are symmetric and PSD, then $AB$ is PSD if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "68",
"answer_count": 4,
"answer_id": 0
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Why is there no continuous square root function on $\mathbb{C}$? I know that what taking square roots for reals, we can choose the standard square root in such a way that the square root function is continuous, with respect to the metric.
Why is that not the case over $\mathbb{C}$, with respect the the $\mathbb{R}^2$ m... | Here is a proof for those who know a little complex function theory.
Suppose $(f(z))^2=z$ for some continuous $f$.
By the implicit function theorem, $f(z)$ is complex differentiable (=holomorphic) for all $z\neq0$ in $\mathbb C$.
However since $f$ is continuous at $0$, it is also differentiable there thanks to Rieman... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How can I evaluate an expression like $\sin(3\pi/2)$ on a calculator and get an answer in terms of $\pi$? I have an expression like this that I need to evaluate:
$$16\sin(2\pi/3)$$
According to my book the answer is $8\sqrt{3}$. However, when I'm using my calculator to get this I get an answer like $13.86$. What I want... | Here’s something I used to tell students that might help. Among the angles that you’re typically expected to know the trig. values for ($30,$ $45,$ $60$ degrees and their cousins in the other quadrants), the only irrational values for the sine, cosine, tangent have the following magnitudes:
$$\frac{\sqrt{2}}{2}, \;\; \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Finding an indefinite integral I have worked through and answered correctly the following question:
$$\int x^2\left(8-x^3\right)^5dx=-\frac{1}{3}\int\left(8-x^3\right)^5\left(-3x^2\right)dx$$
$$=-\frac{1}{3}\times\frac{1}{6}\left(8-x^3\right)^5+c$$
$$=-\frac{1}{18}\left(8-x^3\right)^5+c$$
however I do not fully underst... | You correctly recognised x^2 as "almost" thw derivative of
So put u = (8 - x^3), and find du/dx = -3x^2.
The your integral becomes (-1/3)∫(-3x2)(8−x3)^5dx
= (-1/3) ∫ u^5 (du/dx) dx
= (-1/3) ∫ u^5 du -- which is rather easier to follow. It is the change of variable procedure, which is the reverse of the chain ru... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the proof that covariance matrices are always semi-definite? Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors.
$\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3} \... | A symmetric matrix $C$ of size $n\times n$ is semi-definite if and only if $u^tCu\geqslant0$ for every $n\times1$ (column) vector $u$, where $u^t$ is the $1\times n$ transposed (line) vector. If $C$ is a covariance matrix in the sense that $C=\mathrm E(XX^t)$ for some $n\times 1$ random vector $X$, then the linearity o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
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Find the class of equivalence of a element of a given equivalence relation. Yesterday on my Abstract Algebra course, we were having a problem with equivalence relations. We had a given set: $$A = \{a, b, c\}$$
We found all the partitions of $A$, and one of them was: $$P = \{ \{a\} , \{b, c\} \}$$
Then we built an equiv... | When you have an equivalence relation $R$ on a set $X$, and an element $x\in X$, you can talk about the equivalence class of $x$ (relative to $R$), which is the set $$[x] = \{y\in X\mid (x,y)\in R\} = \{y\in X\mid (y,x)\in R\} = \{y\in X\mid (x,y),(y,x)\in R\}.$$
But I note that your professor did not say "equivalence ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114111",
"timestamp": "2023-03-29T00:00:00",
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} |
Estimating a probability of head of a biased coin The question is: We assume a uniform (0,1) prior for the (unknown) probability of a head. A coin is tossed 100 times with 65 of the tosses turning out heads. What is the probability that the next toss will be head?
Well, the most obvious answer is of course prob = 0.6... | $0.65$ is the maximum-likelihood estimate, but for the problem you describe, it is too simple. For example, if you toss the coin just once and you get a head, then that same rule would say "prob = 1".
Here's one way to get the answer. The prior density is $f(p) = 1$ for $0\le p\le 1$ (that's the density for the unifo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
prove that $g\geq f^2$ The problem is this:
Let $(f_n)$ a sequence in $L^2(\mathbb R)$ and let $f\in L^2(\mathbb R)$ and $g\in L^1(\mathbb R)$. Suppose that $$f_n \rightharpoonup f\;\text{ weakly in } L^2(\mathbb R)$$ and $$f_n^2 \rightharpoonup g\;\text{ weakly in } L^1(\mathbb R).$$ Show that $$f^2\leq g$$ almost eve... | Well it is a property of the weak convergence that every weak convergent sequence is bounded and
$$||f||\leq \lim\inf ||f_{n}||$$
then for every $\Omega \in \mathbb{R}^n$ measurable with finite measure we have
$$\left(\int_\Omega f^2\right)^{\frac{1}{2}}\leq \lim\inf\left(\int_\Omega f_n^2\right)^{\frac{1}{2}}$$
That i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Gentzen's Consistency Proof confusion I am recently finding some confusion. Some texts say that Gentzen's Consistency Proof shows transfinite induction up to $\varepsilon_0$ holds, while other texts say that consistency can be shown up to the numbers less than $\varepsilon_0$, but not $\varepsilon_0$. Which one is corr... | Since $\epsilon_0$ is a limit ordinal when you say induction up to $\epsilon_0$ you mean every ordinal $<\epsilon_0$. In fact the confusion is only understanding in the terminology used, as both mean the same thing.
For example, induction on all the countable ordinals would be just the same as induction up to $\omega_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Product of adjacency matrices I was wondering if there was any meaningful interpertation of the product of two $n\times n$ adjacency matrices of two distinct graphs.
| The dot product of the adjacency matrix with itself is a measure of similarity between nodes. For instance take the non-symmetric directed adjacency matrix A =
1, 0, 1, 0
0, 1, 0, 1
1, 0, 0, 0
1, 0, 1, 0
then the dot of $A^T$A (gram matrix) gives the un-normalized similarity between column i and column j which is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 3,
"answer_id": 2
} |
What are all pairs $(a,b)$ such that if $Ax+By \equiv 0 \pmod n$ then we can conclude $ax+by = 0 \pmod n$? All these are good pairs:
$$(0, 0), (A, B), (2A, 2B), (3A, 3B), \ldots \pmod{n}$$
But are there any other pairs?
actually it was a programming problem with $A,B,n \leq 10000$ but it seems to have a pure solution.
| If $\rm\:c\ |\ A,B,n\:$ cancel $\rm\:c\:$ from $\rm\:Ax + By = nk.\:$ So w.l.o.g. $\rm\:(A,B,n) = 1,\:$ i.e. $\rm\:(A,B)\equiv 1$.
Similarly, restricting to "regular" $\rm\:x,y,\:$ those such that $\rm\:(x,y,n) = 1,\:$ i.e. $\rm\:(x,y)\equiv 1,\:$ yields
Theorem $\rm\:\ If\:\ (A,B)\equiv 1\equiv (x,y)\:\ and\:\ Ax+By... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Calculating the partial derivative of the following I think I may be missing something here,
$$f(x,y)=\left\{ \frac {xy(x^{2}-y^{2})}{x^{2}+y^{2}}\right.\quad(x,y)\neq (0,0)$$
Let $X(s,t)= s\cos(\alpha)+t\sin(\alpha)$ and $Y(s,t)=-s\sin(\alpha)+t\cos(\alpha)$, where $\alpha$ is a constant, and Let $F(s,t)=f(X(s,t), Y(s... | Let $f:\ (x,y)\mapsto f(x,y)$ be an arbitrary function and put $$g(u,v):=f(u\cos\alpha + v\sin\alpha, -u\sin\alpha+v \cos\alpha)\ .$$
Using the abbreviations
$$c:=\cos\alpha, \quad s:=\sin\alpha,\quad \partial_x:={\partial\over\partial x}, \quad\ldots$$
we have (note that $c$ and $s$ are constants)
$$\partial_u=c\part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to pronounce "$\;\setminus\;$" (the symbol for set difference) A question for English speakers. When using (or reading) the symbol $\setminus$ to denote set difference —
$$A\setminus B=\{x\in A|x\notin B\}$$
— how do you pronounce it?
If you please, indicate in a comment on your answer what region you're from (what... | I usually say "A without B," but it depends on my mood that day
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/114488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 10,
"answer_id": 4
} |
Coupon Problem generalized, or Birthday problem backward. I want to solve a variation on the Coupon Collector Problem, or (alternately) a slight variant on the standard Birthday Problem.
I have a slight variant on the standard birthday problem.
In the standard Coupon Collector problem, someone is choosing coupons at ra... | This is a statistical problem, not a probabilistic problem: you have observed data (the value $p$) and seek to infer the underlying probabilistic process (the parameter $k$). The process going from $k$ to $p$ is understood, but the reverse is much more difficult. You cannot "solve" this problem of parameter estimation.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is there a closed-form solution to this linear algebra problem? $A$ and $B$ are non-negative symmetric matrices, whose entries sum to 1.0.
Each of these matrices has $\frac{N^2-N}{2}+N-1$ degrees of freedom.
$D$ is the diagonal matrix defined as follows (in Matlab code):
$$D=\text{diag}(\text{diag}(A*\text{ones}(N)))^{... | The diagonal entries of $D$ are the reciprocals of the row sums of $A$. The row sums of $B$ are those of $A$. Thus $D$ is known. Then $A$ can be obtained as
$$A=\frac1{\sqrt D}\sqrt{\sqrt DB\sqrt D}\frac1{\sqrt D}\;,$$
or, if you prefer,
$$A=D^{-1/2}\left(D^{1/2}BD^{1/2}\right)^{1/2}D^{-1/2}\;.$$
According to this post... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proving Cauchy's Generalized Mean Value Theorem This is an exercise from Stephen Abbott's Understanding Analysis. The hint it gives on how to solve it is not very clear, in my opinion, so I would like for a fresh set of eyes to go over it with me:
pp 143 Exercise 5.3.4. (a) Supply the details for the proof of Cauchy's ... | Note that
$$\begin{eqnarray}h(a)&=&[f(b)-f(a)]g(a)-[g(b)-g(a)]f(a)\\
&=&f(b)g(a)-g(b)f(a)\\
&=&[f(b)-f(a)]g(b)-[g(b)-g(a)]f(b)\\
&=&h(b)\end{eqnarray}$$
and so $h'(c)=0$ for some point $c\in (a,b)$. Then differentiate $h$ normally and note that this makes $c$ the desired point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/114694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
How many combinations of 6 items are possible? I have 6 items and want to know how many combinations are possible in sets of any amount. (no duplicates)
e.g. It's possible to have any of the following:
1,2,3,4,5,6
1,3,5,6,2
1
1,3,4
there cannot be duplicate combinations:
1,2,3,4
4,3,2,1
Edit: for some reason I cannot... | Your are asking the number of subsets of a set with n elements.{1,2,3,...,n}
Each subset can be represented by a binary string, e.g for the set {1,2,3,4,5,6} the string 001101 means the subset that
does not contain the element 1 of the set, because the 1st left character of the string is 0
does not contain the element ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
For $x_1,x_2,x_3\in\mathbb R$ that $x_1+x_2+x_3=0$ show that $\sum_{i=1}^{3}\frac{1}{x^2_i} =({\sum_{i=1}^{3}\frac{1}{x_i}})^2$ Show that if $ x_1,x_2,x_3 \in \mathbb{R}$ , and $x_1+x_2+x_3=0$ , we can say that:
$$\sum_{i=1}^{3}\frac{1}{x^2_i} = \left({\sum_{i=1}^{3}\frac{1}{x_i}}\right)^2.$$
| Hint:
What is value of $\frac{1}{x_1.x_2}+\frac{1}{x_2.x_3}+\frac{1}{x_3.x_1}$ ,when $x_1+x_2+x_3=0$.
If you got the value of $\frac{1}{x_1.x_2}+\frac{1}{x_2.x_3}+\frac{1}{x_3.x_1}$, then proceed by expanding $(\sum_{i=1}^3 \frac{1}{x_i})^2$ by using the formula $(a+b+c)^2= a^2+b^2+c^2+ 2(ab+bc+ac)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/114788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof of a formula involving Euler's totient function: $\varphi (mn) = \varphi (m) \varphi (n) \cdot \frac{d}{\varphi (d)}$ The third formula on the wikipedia page for the Totient function states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$
where $d = \gcd(m,n)$.
How is this claim justi... | You can write $\varphi(n)$ as a product $\varphi(n) = n \prod\limits_{p \mid n} \left( 1 - \frac 1p \right)$ over primes.
Using this identity, we have
$$
\varphi(mn)
= mn \prod_{p \mid mn} \left( 1 - \frac 1p \right)
= mn \frac{\prod_{p \mid m} \left( 1 - \frac 1p \right) \prod_{p \mid n} \left( 1 - \frac 1p \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
"answer_id": 2
} |
Rationality test for a rational power of a rational It has been known since Pythagoras that 2^(1/2) is irrational. It is also obvious that 4^(1/2) is rational. There is also a fun proof that even the power of two irrational numbers can be rational.
Can you, in general, compute whether the power of two rational numbers ... | We can do this much quicker than using prime factorization. Below I show how to reduce the problem to testing if an integer is a (specific) perfect power - i.e. an integer perfect power test.
Lemma $\ $ If $\rm\,R\,$ and $\,\rm K/N\:$ are rationals, $\rm\:K,N\in\mathbb Z,\ \gcd(K,N)=1,\,$ then $$\rm\:R^{K/N}\in\Bbb ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
How to show that these two random number generating methods are equivalent? Let $U$, $U_1$ and $U_2$ be independent uniform random numbers between 0 and 1. Can we show that generating random number $X$ by $X = \sqrt{U}$ and $X = \max(U_1,U_2)$ are equivalent?
| For every $x$ in $(0,1)$, $\mathrm P(\max\{U_1,U_2\}\leqslant x)=\mathrm P(U_1\leqslant x)\cdot\mathrm P(U_2\leqslant x)=x\cdot x=x^2$ and $\mathrm P(\sqrt{U}\leqslant x)=\mathrm P(U\leqslant x^2)=x^2$ hence $\max\{U_1,U_2\}$ and $\sqrt{U}$ follow the same distribution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/114950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Estimate the sample deviation in one pass We've learned this algorithm in class but I'm not sure I've fully understood the correctness of this approach.
It is known as Welford's algorithm for the sum of squares, as described in: Welford, B.P., "Note on a Method for Calculating Corrected Sums of Squares and Products", T... | $v_n$ is going to be $\sum_{i=1}^n (Y_i - \overline{Y}_n)^2$ where $\overline{Y}_n = \frac{1}{n} \sum_{i=1}^n Y_i$. Note that by expanding out the square,
$v_n = \sum_{i=1}^n Y_i^2 - \frac{1}{n} \left(\sum_{i=1}^n Y_i\right)^2$.
In terms of $m_k = \sum_{i=1}^k Y_i$, we have
$$v_n = \sum_{i=1}^n Y_i^2 - \frac{1}{n} m_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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