Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Determining the sampling distribution Let the random variable $X$ represent the number of defective components in
a lot of components. Assume that $X$ can take on four values: $0, 1, 2, 3$. The probability distribution of $X$ is shown in the table below:
X | 0 | 1 | 2 | 3
P(X) | 0.4 | 0.2 | 0.1 | 0.3
1) Ra... | The possible values of the average number are $0$, $1/2$, and so on up to $3$, a total of seven possible values. If the random variable $Y$ is the average number, we want to compute $\Pr(Y=y)$ for these seven possible values.
For example the probability that $Y=1$ is the probability the sum is $2$, which can happen i... | {
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"url": "https://math.stackexchange.com/questions/1709542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is a function differentiable in $x$ if $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\infty$? All the definitions of differentiability I found (Wolfram Mathworld for instance) only require this limit to exist, but say nothing about the domain in which that has to happen.
So what if that limit is $\pm\infty$? Wouldn't (... | This is an unfortunate case of terminology getting the better of us. The definition of a limit is
The limit of the function $f(x)$ as $x$ approaches $a$ exists if there is a number $L$ such that for all $\epsilon>0$, there exists a $\delta>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$. In this case, we sa... | {
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How do I prove that "If prime p does not divide natural number m, then gcd(p,m) = 1" I am having a problem with this.
If prime p does not divide natural number m, then gcd(p,m) = 1
I had to use this for my another proof and because I thought it was quite intuitive, I just assumed this is true and used it for my proof ... | Your proof might be different depending on your definition of GCD. The definition I prefer is the following: a GCD of two integers $m,p$ is a number $d$ which divides $m$ and $p$, and which has the property that if $e$ is an integer dividing $m$ and $p$, then $e$ divides $d$. A GCD of two nonzero integers exists, is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $f\big(f(x)\big)$ has at least as many real roots as $f(x)$ $f(x)$ is a real polynomial of odd degree. Show that $f\big(f(x)\big)$ has at least as many real distinct real roots, counted without multiplicity, as $f(x)$.
| Since $f$ is of odd degree,
$\lim_{x \to \infty} f(x)
=\pm \infty
$
depending on the sign
of the highest order term,
and
$\lim_{x \to -\infty} f(x)
=\mp \infty
$,
with the other sign being used.
Therefore
$f(x)$
takes the reals onto the reals
(with possible multiple occurrances
where $f$ is not monotonic),
so
$f(f(x))$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709837",
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"source": "stackexchange",
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$G$ is cyclic $iff$ there is an element of order $|G|$. I'm trying to prove that if there is an element of order $|G|=n$ there the group is cyclic. But I have some problem. Let $a\in G\backslash \{1\}$. And consider $\left<a\right>$ which is cyclic. Let $b\in G$. Then, $b^n=1=a^n\in \left<a\right>$. Does the fact that ... | By definition $G$ is cyclic iff $G=\langle a\rangle$ and so order of $a$ should be $|G|$. And if the order of $a$ is $|G|=n$ then $a$ generate $n$ distinct elements, and so $a$ generate whole of $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Doubt in the solution of differential equation I know that the solution to the differential equation $\frac{dy}{dt}=y(a-by), a>0, b>0, y(0)=y_0$ can be derived using integration using partial fractions, and the final result is: $$y=a/(b+ke^{-at})$$ where $k$ is $(a/y_0)-b$.
My doubt is: Say $y_0<0$. Then the differenti... | It is a classical differential equation called the logistic differential equation http://math.usu.edu/~powell/biomath/mlab3-02/node2.html
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove there are no prime numbers in the sequence $a_n=10017,100117,1001117,10011117, \dots$ Define a sequence as $a_n=10017,100117,1001117,10011117$. (The $nth$ term has $n$ ones after the two zeroes.)
I conjecture that there are no prime numbers in the sequence. I used wolfram to find the first few factorisations:
$10... | Another way to find the inductive relationship already cited, from a character manipulation point of view:
Consider any number in the sequence, $a_n$. To create the next number, you must:
*
*Subtract $17$, leaving a number terminating in two zeroes;
*Divide by $10$, dropping one of the terminal zeroes;
*Add $1$, ... | {
"language": "en",
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Let $A$ be a complex matrix and $AA^t=A^2$, is it always right that $A$ is symmetric? It is a well-known result that when $A$ is a real matrix and $AA^t=A^2$ then $A$ is symmetric. I guess the same proposition is right for a complex matrix, but I can't prove it. Could someone give me some hints on this problem? I will ... | This is not true in $M_n(\mathbb{C})$, $n\geq 3$.
A very simple counterxample is the following:
Let $v^t=(0,0,1,0,...,0)$ and $w^t=(1,i,0,...,0)$.
Notice that $w^tw=1+i^2=0$ and $w^tv=0$. Define $A=vw^t$.
Now, $A^2=(w^tv)A=0$ and $AA^t=(w^tw)vv^t=0$, but $A=vw^t\neq w^tv=A^t$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find equation for mass in gravity A satellite is moving in circular motion round a planet.
From the physics we know that
$$\Sigma F_r = ma_r = \frac{GMm}{r^2}$$
So I wanted to find the equation for $M$ knowing also that
$$v = \omega r = \frac{2\pi r}{T}$$
and
$$a_r = \frac{v^2}{r}$$
Thus,
$$ma_r = \frac{GMm}{r^2}$$
$$... | $\frac{4\pi^2r^2}{T^2}/r=\frac{4\pi r}{T^2}$ and G should go down not up in numerator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1710340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine whether $p_n$ is decreasing or increasing, if $p_{n+1} = \frac{p_n}{2} + \frac{1}{p_n}$ If $p_1 = 2$ and $p_{n+1} = \frac{p_n}{2}+ \frac{1}{p_n}$, determine $p_n$ is decreasing or increasing.
Here are the first few terms:
$$p_2 = \frac{3}{2}, p_3 = \frac{3}{4} + \frac{2}{3} = \frac{17}{12}, p_4 = \frac{17}{24... | AM/GM. $\frac{x}{2}+\frac{1}{x}<x$ iff $x^2>2$. So it is enough to show that if $x^2>2$ then $(\frac{x}{2}+\frac{1}{x})^2>2$ or $\frac{x^2}{4}+\frac{1}{x^2}>1$. But by AM/GM $(\frac{x^2}{4}+\frac{1}{x^2})/2>\sqrt{\frac{1}{4}}$.
| {
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B-splines locally controlled I have read that in contrast of the thin-plate splines, B-splines are locally controlled, which makes them computationally efficient even for large number of control points.
I didn't understand what does it mean by saying that B-splines are locally controlled. Can anyone help with this?
| It means that value of B-spline at a point depends just on few control points localized nearby. And vice versa, if you modify a control point, or coefficient of one basisfunction it will affect just some local nighborhood.
In case of cubic B-spline (which is most common e.g. in computer graphics ) in 1D value $f(x)$ a... | {
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what is the relation between $f(x+1)$ and $f(x)$? I searched so much over math sites and google but I didn't find helpful hints and required knowledge or the specific name of this topic in function.
I stuck in relation and operations on $f(x)$ which is really important for solving problems.
for example relation betwee... | One can view $f$ as a machine that takes an input (usually called $x$) and outputs something based on that input.
It is usually written in the general case as something like $f(x)=3x-5$ which says whatever our input $x$ happens to be, we output something that is (in this case) $3$ times that input and then subtract fiv... | {
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Finding a limit without using L'Hospital's rule I encountered an issue for finding the limit for a function, one small step I just can't get.
I know that this is classic case for L'Hospital's however our calculus course didn't reach it and I can not use it for this one.
$
2.3 .m,n\in\mathbb{N}\quad\underset{x\rightarr... | $\lim_{x\to 1} \dfrac{x^m - 1}{x^n - 1} =
\lim_{x\to 1} \dfrac{(x-1)(x^{m - 1} + x^{m - 2} + x^{m - 3} + ... + 1)}{(x-1)(x^{n - 1} + x^{n - 2} + x^{n - 3} + ... + 1)} =
\lim_{x\to 1} \dfrac{(x-1)\sum_{i=1}^{m} x^{m-i}}{(x-1)\sum_{j=1}^{n} x^{n-j}} =
\lim_{x\to 1} \dfrac{\sum_{i=1}^{m} x^{m-i}}{\sum_{j=1}^{n} x^{n-j}} =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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An exercise on indicator function in a measure space Given $(\Omega, \mathbb F, \mu)$ a measure space, $(A_n)$ a sequence of measurable sets, $f: (\Omega, \mathbb F) \to (\mathbb R, \mathbb B(\mathbb R))$ an integrable function such that: $\displaystyle \lim_{n\to\infty}\int_\Omega |1_{A_n} - f|d\mu = 0$, where $1_{A_n... | Yes, your proof is correct.
Alternatively, we can notice that the assumption is $\sum_{n\geqslant 0}\lVert \mathbf 1_{A_n}-\mathbf 1_A\rVert_1\lt \infty$. We can use the fact that if $\sum_{n\geqslant 1}\lVert f_n\rVert_1\lt \infty$, then
$\sum_{n\geqslant 1}\lvert f_n\rvert\lt \infty$ almost everywhere hence $f_n\to... | {
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How do I prove $2^0 + 2^1 + 2^2 + 2^3 +\cdots + 2^{d-1} \le n - 1$ $\space$ if $\space$ $d = \lfloor \log_2 n \rfloor$? I was given this inequality in university, me and my friends can't solve it, hope someone here can explain me:
$2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^{d-1} \le n - 1$ $\space$ if $\space$ $d = \lfloor \l... | Observe that the series $\sum_{k=0}^d 2^k$ is a geometric series which has the closed form $(2^{d}-1)/(2-1) = 2^{d}-1$.
Because $d = \lfloor \log_2n \rfloor$, we have
$$d = \lfloor \log_2n \rfloor \leq \log_2n$$
by definition of the floor function. Raising both sides of the inequality by 2 and subtracting 1 thereafter,... | {
"language": "en",
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Can we always find homotopy of two paths which lies "between" the paths? Let $\gamma_0,\gamma_1:[0,1]\to\mathbb{R}^2$ be paths such that $\gamma_0(0)=\gamma_1(0)$ and $\gamma_0(1)=\gamma_1(1)$. I wish to show that there is a homotopy $\Gamma:[0,1]\times[0,1]\to\mathbb{R}^2$ from $\gamma_0$ to $\gamma_1$ that satisfies... | If you allow the contour of the union of those two paths to form a simple closed curve*, this is true, and can be seen as a consequence of the Schoenflies Theorem together with the fact that the disk is contractible.
*I think this is the case, since you are talking about "unbounded face".
| {
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"timestamp": "2023-03-29T00:00:00",
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Let F be a finite field of characteristic $p$. Show $f(a) = a^p$ is a ring homomorphism, injective, and surjective Let F be a finite field of characteristic $p$. Show that the function $f:F \to F$ defined by $f(a) = a^p$ is a) a ring homomorphism, b) injective and, c) surjective.
I tried to approach this problem by pr... | Hint: a field homomorphism is always injective, and an injective map from a finite set to itself is always surjective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1711175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Computing conditional probability with proxy variable Suppose $P(X | Y)$.
If I only know $P(X | Z)$ and $P(Z | Y)$, can I do
$$
P(X|Y) = P(X|Z)P(Z|Y)?
$$
| We toss a fair coin independently three times. Let $X$ be the event head on the first toss, $Y$ the event head on the second, and $Z$ the event head on the third.
Then $\Pr(X\mid Y)=1/2$ and $\Pr(X\mid Z)\Pr(Z\mid Y)=(1/2)(1/2)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find $f'(1)$ if $f$ is continuous and such that $f (f (x))=1+x$ for every $x$
If $ f $ is a continuous function satisfying $f (f (x))=1+x$ find $f'(1)$.
I just guessed $f (x)=x+1/2$.Any formal method for this sum ?
| The relationship $f(f(x))=x+1$ implies that $f^{-1}(x+1)=f(x)$. This implies that when the inverse function is shifted to the left by one unit it must give back the original function. Indeed, $f(x)=x+1/2$ satisfies this requirement and has $f^{-1}(x)=x-\frac{1}{2}$. Although this doesn't show that this is the only func... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that every power of 6 ends in 6? Yesterday I had the traditional math matriculation exam, and in it there was a question "In what digit does the number $2016^{2016}$ end in?" After the test The Matriculation Examination Board published a pdf in which they show how to basically solve all the problems in the... | Use induction in order to complete the hint given by @ThePortakal.
First, show that this is true for $n=1$:
$6^1=6$
Second, assume that this is true for $n$:
$6^n=10k+6$
Third, prove that this is true for $n+1$:
$6^{n+1}=$
$6\cdot\color{red}{6^n}=$
$6\cdot(\color{red}{10k+6})=$
$60k+36=$
$60k+30+6=$
$10(6k+3)+6$
Plea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is continuous function finite-valued in $R^n$? Let $f$ be a continuous function defined on $\mathbb{R}^n$. The range of $f$ is in the extended real numbers. Is $f<\infty \ \forall x\in \mathbb{R}^n$ ? And why?
| In the "usual" topology on the extended real numbers $[-\infty, \infty]$, a neighborhood of $\infty$ is a set containing some interval $(a, \infty] = (a, \infty) \cup \{\infty\}$ with $a$ real, and similarly for neighborhoods of $-\infty$.
If that's true in your setting, $[-\infty, \infty]$ is homeomorphic to a closed,... | {
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"timestamp": "2023-03-29T00:00:00",
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Simple question about discrete metric and openness. You may think this is silly question, but I'm really confused.
In discrete metric, every singleton is an open set.
And, the proof goes like this
$\forall x \in X$, by choosing $\epsilon < 1$, $N_\epsilon(x) \subset ${$x$}
However, if discrete metric is defined in $X$,... | A metric on $X$ is a mapping $d:X \times X \to [0,\infty[$, so you can take $d = 1/2$ even if no pairs with distance $1/2$ exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1711719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to result in moment generating function of Weibull distribution? So I'm trying to get the Weibull distribution moment generating function
$$\sum_{n=0}^\infty \frac{t^n \lambda^n}{n!} \Gamma(1+n/k)$$
(which can be found here https://en.wikipedia.org/wiki/Weibull_distribution)
I'm trying to do it with the definition... | $$\begin{align}
M(t) &= E(e^{tX}) = \int_0^\infty
e^{tx}\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k} dx
&\\ &= \int_0^\infty e^{\lambda tu} ku^{k-1} e^{-u^k} du
&\qquad(\text{with }u=x/\lambda\text{, for }\lambda>0)
\\ &= \int_0^\infty e^{\lambda tx^{1/k}} e^{-x} dx
&\qquad(\text{with }x=u^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1711818",
"timestamp": "2023-03-29T00:00:00",
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Factorisation of a polynomial of degree 5 using limited theorems I have the polynomials $p(X)=X^{5}+P_4X^{4}+P_3X^{3}+P_2X^{2}+P_1X+P_0 \in\mathbb{Z_2[x]}$
I need to determine all $p(x)$ that can be factored into irreducible polynomials of degree three and two.
The catch is that I cannot use most of the techniques usua... | Use the no roots criterion to determine all irreducible quadratics and cubics. Quadratics are very simple, there is only $x^2+x+1$. Cubics are a little more work. But not much. The cubic has to have shape $x^3+ax^2+bx+1$ where there is an odd number of $1$'s. Once you have your list of cubics, multiply each by $x^2+x+1... | {
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Is it really true that "if a function is discontinuous, automatically, it's not differentiable"? I while back, my calculus teacher said something that I find very bothersome. I didn't have time to clarify, but he said:
If a function is discontinuous, automatically, it's not differentiable.
I find this bothersome beca... | Flagrantly ignoring your specific example: suppose a function $f$ is differentiable at a point $x$. Then by definition of differentiability:
$$\lim_{h\rightarrow0}\frac{f(x+h) - f(x)}{h}$$
must exist (and by this notation I mean the limits exist in both the positive and negative directions and are equal). Since the bot... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$ For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite place of $K$ is either an embedding of $K$ ... | Expanding what I believe Franz Lemmermeyer meant with his hint. This is closely built into all the other answers as well (+1 to y'all), but I learned a trick from reading all this, so I can't keep the lid on...
Assume contrariwise that there exist a non-trivial triple $(a,b,c)$ of integers such that
$$
|(1-\sqrt2)^a(2-... | {
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Is there a difference between $\sin^2(x)$ and $\sin(x)^2$? Is there a difference between these? Or are they the same?
If they're the same, then why does $\sin^2(x)$ seem like its used more often?
| The two possibilities are
$$
P_1 = \big(\sin x\big)^2
\\
P_2 = \sin\big(x^2\big)
$$
The convention (which confuses beginners) is
$$
\sin^2 x = P_1
$$
If you write
$$
\sin x^2
$$
or even
$$
\sin(x)^2
$$
we cannot tell which you mean.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Isomorphism of $k$-Algebras Let $k$ be a field an $A$ a finitely generated $k$-algebra (think of the polynomial ring).
Suppose I have an isomorphism of $k$-algebras
$$
A \cong A/I
$$
for some ideal $I$. If $A$ were a finitely generated $k$-vector space we can conclude $I=0$, as opposed to the non finitely generated cas... | The answer is yes for finitely generated commutative $k$-algebras (Mariano's answer deals with the non-commutative case). More generally, if $A$ is a Noetherian ring and there is an isomorphism $f\colon A/I \to A$, then $I = (0)$.
Why? Let $\pi\colon A\to A/I$ be the quotient map. If $I\neq (0)$, then $f^{-1}[I]$ is a ... | {
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"answer_id": 1
} |
sum of the series $\frac{2^n+3^n}{6^n}$ from $n=1$ to $\infty$ Find the sum of the series $\sum_{n=1}^{\infty} \frac{2^n+3^n}{6^n}=?$
My thoughts: find $\sum_{n=1}^{\infty} 2^n$, $\sum_{n=1}^{\infty} 3^n$ and $\sum_{n=1}^{\infty} 6^n$ (although I don't know how yet...)
Then, $\sum_{n=1}^{\infty} \frac{2^n+3^n}{6^n}= \f... | $$\sum_{i=1}^\infty \frac{2^n + 3^n}{6^n}=\sum_{i=1}^\infty (\frac{2}{6})^n + \sum_{i=1}^\infty (\frac{3}{6})^n=\frac{1}{2}+1=\frac{3}{2}$$
sums from forumla of geometric series $1/3^n$ and $1/2^n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove: $\csc a +\cot a = \cot\frac{a}{2}$
Prove: $$\csc a + \cot a = \cot\frac{a}{2}$$
All I have right now, from trig identities, is
$$\frac{1}{\sin a} + \frac{1}{\tan a} = \frac{1}{\tan(a/2)}$$
Where do I go from there?
| Another solution,
$${1\over \sin(a)}= {1\over \tan(a/2)}-{{1-\tan^2(a/2)}\over {2\tan(a/2)}}$$
$${1\over \sin(a)}={1+\tan^2(a/2) \over 2\tan(a/2)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
} |
Boundary value problem in complex plane, how to find general solution?
Hi
I was trying to find out general solution, can anyone explain how to find it?
| The domain is conformally equivalent to the half-plane, and hopefully you have discussed in class how to solve the Dirichlet problem in the half-plane using the Poisson kernel. Since the Laplace equation is conformally invariant it suffices to determine this conformal map. This can be done by translating the "corner" o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is "Let $M$ is a connected n-manifold, then: $M$ is $R$-orientable iff $H_n(M;R)\cong R$" true? I heard that every topological $n$-manifold $M$ is $\mathbb{F}_2$-orientable, but then for $M=\mathbb{R}^2$ is must be $H_2(\mathbb{R}^2;\mathbb{F}_2)\neq 0$?
In lecture we had the lemma: Let $M$ is a connected n-manifold, ... | Yes, this is true only for compact manifolds (I assume that your "manifolds" are not allowed to have boundary). In fact, $H_n(M;R)\cong R$ iff $M$ is $R$-orientable and compact. You should be able to find a proof in any text that covers Poincaré duality. For instance, this follows from Theorem 3.26 and Proposition 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$R^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group Let $n>1$. I need to show that the space $X=\mathbb{R}^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group.
This is an exercise about the Van Kampen theorem. Certainly, we should prove it by contradiction, but I do not kn... | Fundamental group of a topological group is abelian always. And it's easy to show the given space has non-abelian fundamental group.
Proof of the statement in bold:
Let $a$ and $b$ be two loops in a topological group $(G,\bullet )$ starting at the identity element $e$. We need to show $ a\ast b \simeq b\ast a$, where "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability Problem - Finding a pdf Below is a problem I did. However, I did not come up with the answer in book.
I am thinking that I might have the wrong limits for the integral. I am hoping
somebody can point out what I did wrong.
Bob
Problem
Let $Y = \sin X$, where $X$ is uniformly distributed over $(0, 2 \pi)$. Fi... | your answer is correct indeed just a little mistake. in the first step:
$$P(Y \le y) = P( \sin x \le y ) \Rightarrow P(x_1 \le x \le x_2 )$$ as shown in figure below (sorry figure is for $y=\sin(x+\theta)$ but still is useful .just draw the figure for $\sin(x)$ in your imagination. also figure is from "Probability, Ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1712936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Which convergence test to use I need to prove that the following converges:
$$\sum\limits_{n=0}^{\infty} n^3e^{-n^4}$$
I'm not sure which test to use. I've tried ratio test but it gets messy. I don't think I have any sums to use in the comparison test and the integral test obviously wouldn't work. Thanks
| Directly $\;n-$th root test:
$$\sqrt[n]{n^3e^{-n^4}}=\frac{\left(\sqrt[n]n\right)^3}{e^{n^3}}\xrightarrow[n\to\infty]{}0<1$$
and thus the series converges
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Find the volume of the shapes defined as: $n$-dimensional cone
Can this be done without using spherical coordinates?
$$i)\ \frac{x_1^2}{a_1^2}+\cdots+\frac{x_{n-1}^2}{a_{n-1}^2}=\frac{x_n^2}{a_n^2}, \quad 0\leq x_n \leq a_n.$$
$$ii)\ \text{The shape bounded by the following:} \\ \sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}}=... | It can be, but it's usually not because the equations are really messy. Almost no problem is only solvable in a given coordinate system, it's just a question of which integrals are less awful
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
a one-to-one and onto function that cannot be continuous I'm reading Torchinsky's Real Analysis, and am having difficulty demonstrating if
$$\varphi : [0,1]^2 \rightarrow [0,1]$$ is one-to-one and onto, then it cannot be continuous. (5.13)
Unable to demonstrate this myself, I did some research and found what seems to... | We assumed the existence of such a function $\varphi$, did some work to arrive at a contradiction (in particular, the existence of a continuous map that maps a connected set to a set that is not connected). Therefore the assumption was not consistent.
The author might have meant $\varphi$ instead of $\varphi^{-1}$. Tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the stationary point of $f(x,y)=\sin x \sin y \sin (x+y)$ How to find the stationary point(s) of $f(x,y)=\sin x \sin y \sin (x+y)$
With $x,y\in(0,\pi)$
So far I have found $$\nabla f =(\color{red}{\sin x\cos (x+y)+\cos x\sin (x+y)\sin y},\color{blue}{\sin y\cos (x+y)+\cos y\sin (x+y)\sin x)}$$
So we need
$... | $\left(\sin x \cos(x+y) + \cos x\sin(x+y)\right) \sin y = 0$ when $\sin x \cos(x+y) + \cos x\sin(x+y) = 0$ or when $\sin y = 0$. I trust that you know when $\sin y = 0$. What about that first one?
\begin{align}
\sin x \cos(x+y) + \cos x\sin(x+y) &= 0 \\
\sin x \cos(x+y) &= -\cos x\sin(x+y)\\
-\tan x &= \tan(x+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How is it posible that $f + g \in O(f)$? I am confused how to do this question. Intuitively it doesn't even make sense how a function $f$ plus another function is in $O(f)$. How can I approach this question:
$$
n\log(n^7)+n^{\frac{7}{2}} \in O(n^{\frac{7}{2}}).
$$
We know the fact that $\log n < n$ and I tried factori... | The easiest way is to take the limit: $\frac{n^{\frac{7}{2}}+7 n \log n}{n^{\frac{7}{2}}} \to_n 1$, hence these two functions are of the same order.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Problem rearranging a trigonometric expression I've been trying to figure out why this equation is satisfied:
$$\frac{1-(\cos(x))^3}{x^2}=\frac{2\cdot(\sin(\frac{x}{2}))^2}{x^2}\cdot(1+\cos(x)+(\cos(x))^2)$$
but I can't find the proper trigonometric changes in order to change from one to another. I know that the sine c... | One may observe that, in general,
$$
a^3-b^3=(a-b)(a^2+ab+b^2)
$$ giving
$$
1-(\cos x)^3=(1-\cos x)(1+\cos x+(\cos x)^2)
$$ then, by using $2\sin^2(x/2)=1-\cos x$ as noticed by @André Nicolas, you are Ok with your expression.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the Maclaurin series for $\ln(2-x)$ A little unsure if the result I got makes sense, so I want to ask here to be sure I am not doing something very silly.
The Maclaurin series is given by $f(x)=\sum_{n=0}^{\infty}\frac{f^{n}(0)}{n!}x^n$
First I need a formula for the $n$th derivative of $x$. The derivative of $\... | The given answer is not only sensible, it is correct. The "extra" $\ln{2}$ term is part of the solution. Not every function has all its terms in the Maclaurin series fitting a perfect pattern.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Is the tensor product over $B$ of two flat $A$-modules flat over $A$? Given a morphism of commutative rings $A\to B$ such that $B$ is a flat $A$-module and given $M$, $N$ two $B$-modules flat as $A$-modules, is the tensor product $M\otimes_B N$ flat over $A$??
The tensor product $M\otimes_A N$ is flat over $A$, the pro... | Assume that as $B$-modules, you have that $N= B\otimes_A N$: note that this is generally false. There is an obvious map $B\otimes_A N\to N$ that sends $b\otimes n\to bn$. One might be tempted to use the functional inverse $n\to 1\otimes n$, but this is not necessarily $B$-linear, unless say $A\to B$ is onto. If it were... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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How do I find the vector $T\begin{pmatrix} 5 & 0 \\ -10 & -13 \end{pmatrix}$? I defined a function $T: M^R_{2x2} \rightarrow R_4[x]$ and I defined:
$T\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} = x^2$
$T\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} = 3x - 4$
$T\begin{pmatrix} 0 & 2 \\ 4 & 5 \end{pmatrix} = 2x^2 - 7$
Ho... | Hint: You'll need to determine whether there exist $a,b,c$ for which
$$
\pmatrix{5&0\\-10&-13} =
a\pmatrix{2&3\\1&0} +
b\pmatrix{1&0\\0&2} +
c \pmatrix{0&2\\4&5}
$$
and, if such $a,b,c$ exist, find them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine if the function $f$ is even, odd or neither given the graph of $f$ The following picture is a graph of a function $f$. I am to determine whether the function is even, odd, or neither. I reasoned that $f$ is odd, because if the graph is rotated $\pi$ radians, the graph is reproduced perfectly. The graph is cle... | You're right that the graph has a symmetry - if you rotate about a certain point on the $y$-axis, you preserve the graph. Algebraically, it has the property:
$$f(x)=2c-f(-x)$$
where $c=f(0)$ is the intersection of the graph and the $y$-axis. The condition of oddness is more strict, however, it demands:
$$f(x)=-f(-x)$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Showing $u'=v$ a.e. given $u_k \to u$ and $u'_k \to v$ in $L_2(\mathbb{R})$. Suppose $(u_k)$ is a sequence of differentiable functions in $L_2(\mathbb{R})$ satisfying
(1) There is a $u \in L_2(\mathbb{R})$ so that $\| u_k - u\|_2 \to 0$.
(2) There is a $v \in L_2(\mathbb{R})$ so that $\|u'_k - v \|_2 \to 0$.
If $u$ is ... | Define $w(x) := \int_0^x v(t)\,dt$. Then $w$ is differentiable a.e. and $w' = v$ a.e. Let $\varphi\in C_0^\infty(\mathbb R)$. Then
$$
\langle u'-w',\varphi\rangle = \langle w-u,\varphi'\rangle = \langle w-u_k,\varphi'\rangle + \langle u_k-u,\varphi'\rangle.
$$
Now,
$$
\langle w-u_k,\varphi'\rangle = \langle u_k'-v,\var... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1713920",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Find $\frac{y}{x}$ from $3x + 3y = yt = xt + 2.5x$ I need to find the ratio of
$$\frac{y}{x}$$
If given that
$$3x + 3y = yt = xt + 2.5x$$
So what I tried is:
$$t = \frac{3x + 3y}{y}$$
And then put it in the equation
$$\frac{x(3x + 3y)}{y} + 2.5x = \frac{(3x + 3y)}{y}y$$
$$\frac{x(3x + 3y)}{y} + 2.5x = 3x + 3y$$
$$\fra... | Assuming your calculations so far are correct (I didn't check), you are almost there. Divide both sides by $y$, you will get
$$\frac {3x^2} {y^2} + \frac {2.5x} y = 3,$$
a quadratic equation for $\frac xy$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Algebraic Varieties vs Smooth Manifolds There are many posts I have read on that subject which seem unclear for me. My main question (it might be silly) is:
"Every non-singular algebraic variety over $\mathbb{C}$ is a smooth
manifold."
(see: http://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also... | The statements it that if the variety is endowed with the relative topology as a subset of $\Bbb C^n$ then it is a manifold.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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About a Possible Decomposition of a Set $E$ of Infinite Measure Can we decompose $m(E)=\infty$ into $E=\bigcup_{n\in \mathbb{Z}}E_n$ where $E_n=E\bigcap[n-1,n]$ where $n \in \mathbb{Z}$? Then we have $\{E_n\}_{n\in \mathbb{Z}}$, a countable collection of measurable subsets of $E$ and of finite measure since $E_n \subs... | *
*First question: Yes. From $E_n \triangleq E \cap [n-1,n]$, it's trivial that $E=\bigcup_{n\in \Bbb Z} E_n$.
*Second question: No. Consider $E = \Bbb R$, then $m(E) = \infty$. $$E_n \triangleq E \cap [n-1,n] = [n-1,n],$$ so "each pair of $E_n$'s is not necessarily disjoint" (as $E_n \cap E_{n+1} = \{n\}$). $$F_n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Mean value theorem question. Show that if $f(2) = -2$ and $f ' (x) \geq 5$ for $x > 2$, then $f(4) \geq 8$.
I know that the $b$ value is $4$ and the $a$ value is $2$ for the mean value theorem equation.
$$f'(c)=\frac{f(4) - f(2)}{4-2}$$
We know the value of $f(2)$ but for $f(4)$ I'm confused there and with the $f ' (x... | Your equation can be rewritten as
$$f(4)=f(2)+(4-2)f'(c).\tag{1}$$
We also know that $c$ is between $2$ and $4$, and therefore in particular $c\gt 2$.
So $f'(c)\ge 5$. Also, $f(2)=-2$. It follows from (1) that
$$f(4)\ge -2+(4-2)(5).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Computing degrees of projective varieties via Chern classes I know that the degree of a projective hypersurface $H \subset \mathbb{P}^n$ can be computed in terms of the Chern class of the normal (line) bundle of $H$. Is there a similar formula for the degree of a higher codimension projective variety in terms of Chern ... | Here is a general answer in terms of Chow rings.
Let $i:Y\hookrightarrow \mathbb P^n$ be a smooth closed subvariety of codimension $r$ and degree $d$, so that for the corresponding cycle class we have $[Y]=dH^r\in A^r(\mathbb P^n)=\mathbb Z\cdot H^r$ .
We have (Hartshorne, page431): $$c_r(N_{Y/X})=i^\ast [Y]=i^\ast... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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How to find the eigenvector when there are multiple instances of the eigenvalues $$\begin{pmatrix}0&1&-1\\2&1&-2\\-1&-1&0\end{pmatrix}$$
The characteristic polynomial is $$\lambda^3-\lambda^2-5\lambda-3=(\lambda+1)(\lambda^2-2\lambda-3)=(\lambda+1)(\lambda+1)(\lambda-3)$$
So I have two eigenvectors with the same eigenv... | When there is an eigenvalue of multiplicity $k>1$, there is an eigenspace of dimension at most $k$. If $k=1$, the dimension is always $1$ as there is always at least one eigenvector $\vec u$, thus all $t\vec u$ are also eigenvectors, thus an eigenspace of dimension $1$.
When $k>1$, it can happen that there are not "eno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Understanding the unit tangent vector The vector $\dot x (s) $ is called the unit tangent vector to the oriented curve $x=x(s)$.
I am told that $x=x(s)$ is a natural representation of a regular curve C.
What does natural representation mean?
The derivative $\dot x(s)=\frac{dx}{ds}$ is defined as the tangent directio... | $s$ is the arclength. Namely, (and we just work in $\mathbb R^2$ here):
if $\gamma:[0,1]\to \mathbb R^2$, is a paramterization of your curve $C$, then
$\gamma (t)=(x(t),y(t))\in C$ and $\gamma(0)=(x_0,y_0); \ \gamma(1)=(x_1,y_1)$
then the length of $C$ from $(x_0,y_0)$ to $(x_1,y_1)$ is given by
$l=\int_{0}^{1}\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the equation of the normal line with the minimum y-coordinate is $ y = \frac{-\sqrt{2}}{2}x + {1\over k}$
Question: The curve in the figure is the parabola $y=kx^2$ where $k>0$.
Several normal lines to this parabola are also shown. Consider the points in the first quadrant from which the normal lines are dr... | You did almost all the work. From
$$ x={-\frac{1}{2kx_0}\pm { (2kx_0 + \frac{1}{2kx_0})} \over 2k}$$
the $x$ you are looking for is the one with the '$-$':
$$x={-\frac{1}{2kx_0}- { (2kx_0 + \frac{1}{2kx_0})} \over 2k}=-\frac{1}{2k^2x_0}-x_0$$
(the other one is simply $x=x_0$).
Now
$$y=k\left(-\frac{1}{2k^2x_0}-x_0\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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How can I mentally calculate $\cos(x), x∈(0.7, 1.2)$ I'm trying to learn how to calculate trig functions in my head. I'm planning on learning $\cos(x), x∈[0,π/2]$ and then using symmetry to calculate the others.
I think the quadratic Maclaurin series at $0$ and the linear at $π/2$ could be calculated in a matter of sec... | Here's a quick approximation. The second-order Taylor series around $x=\pi/3$ is
$$T_2(x) = \frac12 - \frac{\sqrt3}2\left(x-\frac\pi3\right)-\frac14\left(x-\frac\pi3\right)^2.$$
Now, $\frac{\sqrt3}2 = 0.866... \approx \frac{13}{15}$, and $\frac\pi3 \approx 1.05$, so we have
$$T_2(x) \approx \frac12 - \frac{13}{15}(x-1.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1714976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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$a,b,c \in \mathbf{Z}$ such that $a^7+b^7+c^7=45$ Do there exist integers $a,b,c$ such that $a^7+b^7+c^7=45$?
[I have an ugly argument for a negative answer, is it possible to give a "manual" solution?]
| The seventh powers modulo $49$ are $0,\pm 1,\pm 18,\pm 19.$ There is no way to combine three of these to get $45$ modulo $49$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1715085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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What are the principal (different) mechanisms of infinite descent proof? I’m interested in building a list (including, where possible, links to proofs/papers/examples) which presents all known mechanisms of infinite descent (ID).
I think this list would best be presented in two parts:
*
*Answers, each of which outli... | Set theoretic elementary result:
Every subset of a finite set is finite.
(for definition of finite and argument see this math.stackexhange post).
Keep decrementing a positive integer $n$ by $1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$. I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare.
Thanks in advance
| Try to express both $\sqrt{2}$ and $\sqrt[3]{2}$ as rational functions of $a = \sqrt{2}+\sqrt[3]{2}$. The job is simple and easily done via equation $$(a -\sqrt{2})^{3}=2\tag{1}$$ so that $$a^{3}-3\sqrt{2}a^{2}+6a-2\sqrt{2}=2$$ or $$\sqrt{2}=\frac{a^{3}+6a-2}{3a^{2}+2}\tag{2}$$ and we have $$\sqrt[3]{2}=a-\sqrt{2}$$ an... | {
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Where am I violating the rules? Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a seventh-century Indian mathematician) (see here), I considered the function $$\sin \left(\fra... | @Claude Leibivici use the following two point Taylor series in x=-Pi, Pi
$$\frac{z (z-\pi )^3 (z+\pi )^3}{48 \pi ^4}-\frac{5 z (z-\pi )^3 (z+\pi )^3}{16 \pi ^6}+\frac{3 z (z-\pi )^2 (z+\pi )^2}{8 \pi ^4}-\frac{z (z-\pi ) (z+\pi )}{2 \pi ^2}$$ the cuadratic error is superior to any formula above at the same grade
| {
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"timestamp": "2023-03-29T00:00:00",
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Is the autoparallel equation same as the geodesic equation? My question may sound silly but i am self learning differential geometry using watching these lectures (Lecture 8) from 2015 by professor Frederic Schuller.
Can somebody please tell that the auto-parallel equation same as the geodesic equation?
| You cannot talk about geodesics until you have a notion of distance, and curvature is insufficient for that. However, in lecture 10 he introduces the metric tensor, and he shows that the geodesic equation for a given metric takes the form of an autoparallel equation for one specific curvature / connection / covariant d... | {
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How is the binomial expansion of the vectors? I'm trying to find out if there is an attempt to define binomial expansion of vectors. i.e
$$(\overrightarrow a + \overrightarrow b)^n = ?? $$
I tried to google around this (e.g : binomial expansion of vectors), but simple searches do not give any useful keywords as how thi... | You can take a binomial type expansion of the vector expression
$$|a-b|^n,$$
where $|\cdot|$ is the norm. For instance, in $\mathbb{R}^2$ consider $f(x) = |x+y|^n$ Taylor expanded about the point $x = (0, 0)$. For $n=2$ we have
\begin{align}
f(x) & = f(0) + x_1f_{x_1}(0) + x_2f_{x_2}(0) + \frac{1}{2!}\bigg(x_1^2f_{x_1x... | {
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How to show $\text{Im}~ \theta=\Bbb{Q} [\sqrt 2]$ for a homomorphism? How to show $\text{Im} ~\theta=\Bbb{Q} [\sqrt 2]$ for the homomorphism defined as $\theta:\mathbb{Q}[X] \rightarrow \mathbb{R}$ given by $\theta(f(X))=f(\sqrt2)$.
I can show $\Bbb{Q} [\sqrt 2] \subseteq \text{Im}~\theta$ because we can simply take an... | Note that for every $n$: $\sqrt{2}^n\in \mathbb{Q}$ for $n$ even and $\sqrt{2}^n\notin \mathbb{Q}$ for $n$ odd. So, if $f(x)=q_nx^n+q_{n-1}x^{n-1}+\ldots+q_1x+q_0\in \mathbb{Q}[x]$, then: $$f(\sqrt{2})=q_n\sqrt{2}^n+q_{n-1}\sqrt{2}^{n-1}+\ldots+q_1 \sqrt{2}+q_0=\sum_{0\leq i\leq n ~even}q_i\sqrt{2}^{~i}+\sum_{1\leq i\... | {
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Homomorphic image of a free group Given a group $G$ with a generating set $S$, such that $|S|=n$, I need to prove that $G$ is a homomorphic image of $F_n$.
Right now I'm just looking for any tips for how to even start this proof or how to start thinking about the question.
| Hint: $F_n$ is a free object in the category of groups i.e $Hom_{Group}(F_n,G)=Hom_{Set}(\{1,...,n\},G)$.
This is equivalent to saying that the forgetful functor from the category of sets to the category of groups has a left adjoint which associates to a set $S$ the free group $F_S$.
| {
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Unable to prove a simple inequality Let the function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be strictly increasing. That is: $x_2>x_1 \Rightarrow f(x_2)>f(x_1)$.
In the segment $[0;a] \subset \mathbb{Z}$, we have $f(a)<a$
I want to prove that $\forall i \in [0;a] , f(i)<i$ .
I am getting a bit frustrated because it lo... | I don't think there is much of a simpler proof. Certainly every proof needs some kind of method equivalent to induction, be it applying the well-ordering principle or infinite descent.
| {
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Probability of choosing 5 out of 60 in ascending order. The title may be a little misleading. Let's say we choose 5 out of 60 balls. We write down the result which are in a form as $k_1,k_2,k_3,k_4,k_5$.
I have to calculate the probability of this happening :
\begin{aligned}k_1<k_2<k_3<k_4<k_5\end{aligned}
Also, th... | Each choice in which $k_1 < \cdots k_5$ corresponds one-to-one to a way to pick five balls from the set of 60. Prove this statement. Then think about how many ways there are to pick 5 balls from 60.
This should be easy. The second one is a little harder, but having thought about the first one in these terms should help... | {
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Generating Numbers Proof We can do the following actions: multiply by $10$ (add $0$ at the end of number) , multiply by $10$ and add $4$ (add $4$ at the end of number) and divide by $2$. I need to prove that we can get every natural number from $4$ using these actions.
Is it easier to prove that from any number I can ... | This is a trickier problem than I expected, so this is only a partial answer for now.
Working backwards, it's clear that you can get from any number that ends in a $0$ or a $4$ to a smaller number in a single step, simply by deleting the final digit (i.e., divide by $10$, in the first case, or subtract $4$ then divid... | {
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Proving that a countable set in an arbitrary set $ X $ has outer measure zero . It is well known that in $\mathbb{R} $ (well, also in $\mathbb{R}^n$) a countable set $A $ has outer measure zero. It is not really hard to prove and it is a common exercise among the books which I've already done.
However, I'm somewhat stu... | It depends on the (outer) measure and set $X$ you look at. For example let $X = \mathbb{Z}$ and define an outer measure that is not zero on each subset of $\mathbb{Z} $. Or on $\mathbb{R}$ define $\nu(\emptyset) := 0, \nu(A) := 1 \, (\emptyset \neq A \in \mathcal{P}(X))$.
| {
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Let $f : E \rightarrow \mathbb{R}$. Show that if $|f|$ is measurable on $E$ and the set $\{f > 0\}$ is measurable, then $f$ is measurable on $E$. I'm learning about Measure Theory (specifically measurable functions) and need help with the following problem:
Let $f : E \rightarrow \mathbb{R}$. Show that if $|f|$ is mea... | You have that
if $b \geq 0, f^{-1}(]b,a[) = (|f|)^{-1}(]b,a[) \cap \{f > 0 \}$, so it's measurable
if $b < 0, f^{-1}(]b,0]) = (|f|)^{-1}([0,-b[ ) \cap \{f > 0 \}^c$, so it's also measurable
This imply that if $a>0, b<0$, $f^{-1}(]b,a[) = f^{-1}(]0,a[) \cup f^{-1}(]b,0])$ is measurable
if $a < 0, b<a, f^{-1}(]b,a[) = (... | {
"language": "en",
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Prove that every set formed by polynomials of different degrees is linearly independent How to prove that every set formed by polynomials of different degrees is linearly independent.
My problem is in how to make a general set of polynomials of different degrees.
| I would order them in increasing degree. Then, if $\langle S \rangle$ denotes the subspace generated by a subset $S$, I would use the following fact:
FACT: If $u_1, \ldots , u_r$ are linearly independent and $u_{r+1}$ does not belong to $\langle u_1, \ldots, u_r\rangle$, then $u_1, \ldots , u_{r+1}$ are linearly inde... | {
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Roll two dice. What is the probability that one die shows exactly two more than the other die?
Two fair six-sided dice are rolled. What is the probability that one die shows exactly two more than the other die (for example, rolling a $1$ and $3$, or rolling a $6$ and a $4$)?
I know how to calculate the probabilities ... | The probability of rolling a 1 and 3 is 1/18. Same for the probability of 2&4, 3&5, and 4&6.
So the overall probability of the dice being two apart equals 4/18 = 2/9.
| {
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Balls and boxes pigeonhole problem One has $60$ boxes and many (one colored) balls with $8$ different colors. In every box one puts $3$ balls with different colors. Must there exist (at least) two boxes with the same three colored balls?
If one puts $3$ balls with different colors each in a box, we must find first how... | Your explanation is partially correct.
There are $\binom{8}{3} = 56$ ways of selecting three of the eight colors. Since there are $60$ boxes, this means that there must be at least two boxes containing the same combination of three colors.
However, we cannot conclude that there are four boxes containing the same com... | {
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Solve for the values of $x$ in $|x+k|=|x|+k$ where $k$ is a positive real number The question asks me for which values of the real number $x$ is $|x+k|=|x|+k$ where $k$ is a positive real number.
How do I go about this? Can I square both sides to get rid of the absolute value signs? When I do it this way, I get a singl... | If $x\ge0$ and $x+k\ge0$, then the identity holds always. You get $x\ge0$ and $k\ge-x$.
If $x\ge0$ and $x+k<0$, then $-x-k=x+k$, and $x+k=0$ (impossible).
If $x<0$ and $x+k<0$, then $-x-k=-x+k$, and $k=0$ (impossible).
If $x<0$ and $x+k\ge0$, then $x+k=-x+k$, and $x=0$ (impossible).
| {
"language": "en",
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Show that $17$ divides $p-1$ I'm given that $s=2^{17}-1$ and that $p$ is a prime factor of $s$.
First I'm asked to show that $2^{17}\equiv 1(\mod p).$
For this I have simply said that since $p$ divides $s$, this means that $2^{17}-1=pk$ and so $2^{17}\equiv 1(\mod p).$
Next I'm asked to show that $17$ divides $p-1.$
I'... | $2^{17}\equiv 1\mod p$ says the order of $2$ modulo $p$ is a divisor of $17$. However $17$ is prime, and the order of $2$ is not $1$. Hence this order is $17$. From Little Fermat there results $17$ divides $p-1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does the Series $\sum_{n=1}^{\infty} (1-\cos\frac{\pi}{n})$ Converge?
Does the serie $\displaystyle\sum_{n=1}^{\infty}\Bigl(1-\cos\frac{\pi}{n}\Bigr)$ converge?
Limit test of $1-\cos\frac{\pi}{n}$
$$
\lim_{x\to\infty} 1-\cos\frac{\pi}{n} =
1 -\lim_{n\to\infty}\cos\frac{\pi}{n}= 1-1 = 0
$$
I've checked the necessary c... | You can use L'hopital's rule (twice) to show that
$$\lim_{x \rightarrow 0} {1 - \cos x \over x^2} = {1 \over 2}$$
So you can use the limit comparison test with the series ${\displaystyle \sum_{n = 1}^{\infty} {1 \over n^2}}$ to show your series converges.
| {
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How to determine if a sphere is locally isometric to a plane?
Is a sphere of radius one locally isometric to a plane? (Briefly justify your answer.)
How do I go about answering this question. Could someone provide an explanation to what exactly locally isometric means? Also is the radius value of 1 an irrelevant piec... | Hint:
1) We say that a smooth map $F : S_1 →\rightarrow S_2$ between the two surfaces $S_1$,$S_2$ is a local isometry if it preserves distances between two points close to each other.
2) From the Teorema egregium of Gauss we know that the gaussian curvature of a surface is invariant under isometries. And the gaussian c... | {
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Find the point on the cone closest to (1,4,0) Find the point on the cone $z^2=x^2+y^2$ nearest to the point $P(1,4,0)$.
This is a homework problem I've not made much headway on.
| This seems like an exercise in Lagrange Multipliers. You need to minimize the distance function $f(x,y,z) = (x-1)^2 + (y-4)^2 + z^2$ (which is the square distance from $(x,y,z)$ to $(1,4,0)$) subject to the constraint that $g(x,y,z) = x^2 + y^2 - z^2 = 0$. Any point which minimizes such $f$ subject to the constraint wi... | {
"language": "en",
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Geometric intuition of graph Laplacian matrices I am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacian of a graph? For example, why are the eigenvectors of a Laplacian matrix h... | Perhaps the most natural definition is the (oriented) incidence transformation.
The incidence transformation of an edge $(u,v)$ is defined by $E(u,v) = v - u$.
Its adjoint $E^*$ has a dual definition, taking vertices to the sum of incoming edges minus the sum of outgoing ones.
The Laplacian is just $L=EE^*$. Thus, for ... | {
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Find all positive integers n such that $2^2 + 2^5+ 2^n$ is a perfect square. Find all positive integers n such that $2^2 + 2^5 + 2^n$ is a perfect square. Explain your answer.
| We have:
$$2^n+36=m^2$$
$$2^n=(m-6)(m+6)$$
Both $m-6$ and $m+6$ must be powers of $2$ (and note that they differ by $12$). Can you continue?
| {
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What's "$\mathbb Z = (1)$ is cyclic"? As I understand it, $(1)$ is a cyclic group with $1$ being its generator. So, $1^n$ with $n \in \mathbb N$ generates $\mathbb Z^+$ , but what about $\mathbb Z^{-1}$ ? Do we say $(1)$ is a cyclic group with two operations $(+, -)$ defined on it so that $\mathbb Z = (1)$?
| In algebra, when you say that a set $S$ generates some structure $A$, you mean that any element of $A$ can be represented as repeated applications of operations of the structure in question to elements of $S$.
In your case, the structure is a group. A group has not only the binary operation, but also the inverse (and, ... | {
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Proving a trigonometric equation Knowing that : $$ \sin t - \cos t = x $$
Prove that : $$ \cos^3t - \sin^3t = \frac{x}{2}(3-x^2) $$
I tried to solve it by the important corresponding $$ a^3 - b^3 = (a-b)(a²+ab+b²) $$
But I got stuck in the middle and I don't even know if it's correct what I did
| Note that $$a^2=(\sin(t)-\cos(t))^2=\sin^2(t)+\cos^2(t)-2\sin(t)\cos(t)=1-2\sin(t)\cos(t).$$ Hence $$\cos^3(t)-\sin^3(t) = a(\cos^2(t)+\sin^2(t)+\cos(t)\sin(t))=a(1+\frac{1-a^2}{2})=\frac{a}{2}(3-a^2).$$
| {
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Complex analysis: Calculating $\int_{-\infty}^{\infty} \frac{\sin x}{x} dx$ by using $f(z) = \frac{e^{iz} - 1}{iz}$ So I integrate the holomorphic function $\frac{e^{iz} -1}{iz}$ over the half disk in the upper half plane, let me name it $\Gamma$. By using Cauchy's theorem, it equals 0.
$$0 = \int_{\Gamma} \frac{e^{iz}... | $\frac{\cos(t)-1}{it}$ is an odd function so its integral over any $[-R,R]$ is zero.
| {
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Prove that $\mathbb Q[\sqrt[3]2]$ is a field We define the set:
$$\mathbb{Q}[\sqrt[3]2]=\{a_{0}+a_{1}\sqrt[3]{2}+a_{2}\sqrt[3]{2^{2}}:a_{0}, a_1,a_2\in\mathbb{Q}\}$$
It's easy to prove all the properties of fields, except for the unit elements.
So, how can we prove that
$$\forall x\in\mathbb{Q^{*}}[\sqrt[3]2],\exists ... | Doing this for $\sqrt[3]{2}$ is a waste of time. ;-) Exactly because doing it for $\sqrt[n]{2}$ would lead to gigantic computations.
Suppose $r\in\mathbb{C}$ is algebraic over $\mathbb{Q}$. We want to see that the set $\mathbb{Q}[r]$ consisting of all the expressions of the form $a_0+a_1r+\dots+a_nr^n$ is a field.
Let ... | {
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Dimension of Basis of Subspace I was just wondering if there is any "rule" for what the dimension of a basis of a given subspace will be.
For example, a problem I just did involved a vector $v = (1, 2, 3, 4)$ in $\mathbb{R}^4$, and I had to find a basis for the subspace in $\mathbb{R}^4$ consisting of all vectors perpe... |
My intuition for this was to note that the subspace of vectors perpendicular to v is the plane with v as its normal vector. Thus, any two vectors in the plane which are linearly independent would be a basis, and the dimension of the basis would be two.
Unfortunately planes are not always two-dimensional. The correct ... | {
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"timestamp": "2023-03-29T00:00:00",
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What are the limit points of $\sin(2n \pi/3)$? I want to find the limit points of $a_n = \sin(2n \pi/3)$. So far I've identified the subsequences $a_{3n} = 0$ and $a_{1} = \frac{\sqrt{3}}{2}$ and $a_2 = -\frac{\sqrt{3}}{2}$ so are the limit points $\left\{0, \pm \frac{\sqrt{3}}{2}\right\}$? How do I show there aren't m... | A limit point is a point (number) for which there is a subsequence that converges to that point. Now any subsequence must have either infinitely many $0$'s or infinitely many $\dfrac {\sqrt 3} 2$'s or infinitely many $-\dfrac {\sqrt 3} 2$'s, so the limit can ONLY be one of these three numbers.
| {
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Find $x$ in the equation $x^x = n$ for a given $n$ Simply: How do I solve this equation for a given $n \in \mathbb Z$?
$x^x = n$
I mean, of course $2^2=4$ and $3^3=27$ and so on. But I don't understand how to calculate the reverse of this, to get from a given $n$ to $x$.
| Simpler:
If
$x^x = n$,
then $x\ln(x) = \ln(n)
=y$.
Let
$f(x) = x\ln(x)-y
$.
$f'(x)
=\ln(x)+1
$.
Applying Newton's iteration,
starting with $x = \frac{y}{\ln y}$,
$x_{new}
=x-\frac{f(x)}{f'(x)}
=x-\frac{x\ln(x)-y}{\ln(x)+1}
=\frac{x\ln(x)+x-x\ln(x)+y}{\ln(x)+1}
=\frac{x+y}{\ln(x)+1}
$.
Iterate until cooked.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find the remainder of $9^2\cdot 13\cdot 21^2$ when divided by $4$
Find the remainder of $9^2\cdot 13\cdot 21^2$ when divided by $4$
How should I approach this type of questions?
Without calculator of course
I did this:
$9^2\cdot 13\cdot 21^2=81\cdot 13\cdot 441=81\cdot 5733=464,373=33\bmod 4=1 \bmod 4$
| Hint: What is $9\bmod 4$? What is $13\bmod 4$?
How would you proceed from here?
Also, you wrote that $464\,373=1\bmod 4$. Always keep in mind that the "$=$" sign means that two things are exactly the same, in every way. In stead one should write
*
*$\overline{464\,373}=1\bmod 4$,
*$464\,373\bmod 4=1\bmod 4$,
*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Expected value of two successive heads or tails ( I do not understand the answer) Problem: This is problem 33 of section 2.6 (conditioning) in Bertsekas, Tsitsiklis Introduction to Probability 2nd:
We are given that a coin has probability of heads equal to p and tails equal to q and it is tossed successively and indep... | The equation $$E[X\mid H_1,T_2]=1+E[X\mid T_1]$$ reads as follows:
*
*LHS: The expected number of trials after you have tossed a head in the first toss and tails in the second is equal to
*RHS: 1 plus the expected number of tosses given that in the last toss you tossed tails.
The $1+$ in the RHS stands for the firs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A simple binomial identity Is there a simple way of showing that a prime $p$ must divide the binomial coefficient $p^n\choose{k}$ for all $n\geq 1$ and $1\leq k\leq p^n-1$?
| Just a quick remark after the fact: If you accept that $$ (a +b )^{p} \equiv a^p +b^p\pmod p ,$$ for $a$ and $b$ indeterminants,
then
$$(a+b)^{p^n} = \left(\ (a + b )^p\ \right)^{p^{n-1}}\equiv \left(\ a^p + b^p\ \right)^{p^{n-1}}\equiv a^{p^n}+ b^{p^n}\pmod p,$$
which also gives the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Show there in no non constant analytic functions in disk unit s.t $f(z)=f(2z)$ I'm trying to solve the following
Show that there is no non constant analytic function in the unit disc such that $f(z)=f(2z)$.
My try: let $f$ be an analytic function in the unit disk such that $f(z)=f(2z)$.
Now, we can write $\displaysty... | [1]. Another way to show that $a_n=0$ for $n\geq 1$ is that for $0<r<1/2$ and $n\geq 1$ we have $$2 \pi i a_n=(2 \pi i/n!)(d^nf/dz^n)(0)=\int_{|z|=r}f(z)z^{-n-1} \;dz=$$ $$=\int_{|z|=r}f(2 z)z^{-n-1}\;dz=\int_{|z|=r}f(2 z)(2 z)^{-n-1}2^n\;d(2 z)=$$ $$=2^n\int_{|y|=2 r} f(y)y^{-n-1}\;dy= 2^n(2 \pi i/n!)(d^nf/dz^n)(0) =2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
What's the condition for a matrix $A$ ($2N\times 2N$ dimension) to have eigenvalues in pairs $\pm\lambda$?
For a given even dimension square complex matrix $A$ ($2N\times 2N$ dimension),
what's the sufficient and necessary condition for the matrix $A$ such that:
if $\lambda_{1}$ is an eigenvalue, then $\lambda_{2}... | Take any Hermitian matrix of the form $\mathfrak{B}=\left[\begin{array}{c|c}\mathbf{0}_n & A\\\hline A^* & \mathbf{0}_m\end{array}\right],$ where $A$ is $n\times m$. Then if $\lambda$ is an eigenvalue of $\mathfrak{B}$ with multiplicity $k$, then $-\lambda$ is also an eigenvalue with the same multiplicity. This is anot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is there a way to study "how good" is the Newton's Raphson method applied to a function? My question is as simple as that.
When we're applying the fixed point algorithm, we can see if it's going to converge or diverge finding the derivative and checking if the absolute value of that function is less or greater than $1... | As a fixed point iteration
$$
N(x)=x-\frac{f(x)}{f'(x)}
$$
has derivative
$$
N'(x)=\frac{f(x)f''(x)}{f'(x)^2}
$$
and the usual estimates apply. Since at the solution $f(x_*)=0$, you can get intervals around $x_*$ with arbitrarily small contraction constants, compatible with the quadratic convergence speed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1718935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove the convergence of $\int_0^\infty \frac{\ln{x}}{1+x^2}dx$? How to prove the convergence of $\int_0^\infty \frac{\ln{x}}{1+x^2}dx$.Since it's unbounded on both sides, we need to prove the convergence of both $\int_0^1 \frac{\ln{x}}{1+x^2}dx$ and $\int_1^\infty \frac{\ln{x}}{1+x^2}dx$.
$\frac{\ln{x}}{1+x^2}\... | You say that the function is not even non-negative. Have you studied its sign properly on $[0,1]$? What is the matter about its sign?
The function is actually negative on $(0,1]$, so minus the function is positive, and now all comparison tests work just fine.
Since $1+x^2$ lies between $1$ and $2$, we have for $\epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Geometric reasons finite fields have prime power orders? All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
| A finite field is a vector space over some $\mathbb{F}_p$, so it has a basis and its elements are in bijection with some $n$-uples of coordinates in $\mathbb{F}_p$, which tells you it has cardinal $p^n$. I don't know if you consider that a geometric or algebraic argument though.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing that every polynomial over the Algebraic Numbers has a $0$ in the Algebraic Numbers. Let $\mathbb{A}$ denote the field of Algebraic Numbers: the field of all complex numbers that are algebraic over $\mathbb{Q}$. Assuming that every polynomial over $\mathbb{C}$ has a $0$ in $\mathbb{C}$ how would you go about pr... | Let $P\in\mathbb A[X]$. Then there is $a\in\mathbb C$ such that $P(a)=0$. This shows that $a$ is algebraic over $\mathbb A$, so over $\mathbb Q$ (why?). Thus we get $a\in\mathbb A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Gradient of Function I am trying to find the $\nabla F$ with respect to $x_i$ where $F$ is as follows:
$$F(x_0,...,x_n) = c_1\sum_{i=0}^{n}\sum_{j=1}^{k}\frac{1}{||x_i - r_j||_2^2} + c_2\sum_{i=0}^{n-1}||x_{i+1}-x_i||_2^2$$
For clarity I will show my attempts at derivation for each seperate part of the sum.
First part... | The partial derivative $\partial_\alpha = \partial/\partial x_\alpha$ of $F$ is:
\begin{align}
\partial_\alpha F(x_0,\dotsc,x_n)
&=
c_1\sum_{i=0}^{n}\sum_{j=1}^{k}\partial_\alpha \frac{1}{\lVert x_i - r_j\rVert_2^2} +
c_2\sum_{i=0}^{n-1} \partial_\alpha \lVert x_{i+1}-x_i\rVert_2^2 \\
&=
c_1\sum_{i=0}^{n}\sum_{j=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Formula for finding the nth number in this sequence: $[0, 1, 3, 11, 50, 274...]$? The sequence here is what I have discovered to be this: https://oeis.org/A000254, which references Stirling numbers of the First Kind. The formulas provided in that link are very ambiguous with their notation, and I have no idea what to t... | The link you provide has the following formula:
$$a(n)=n!\cdot\sum_{k=0}^{n-1}\frac{1}{n-k}$$
where $a(n)$ is the $n$th number in the series and $n\ge 1$. Thus,
$a(0)=0$
$a(1)=1!\cdot\frac{1}{1-0}=1$
$a(2)=2!\cdot\left(\frac{1}{2-0}+\frac{1}{2-1}\right)=3$
$a(3)=3!\cdot\left(\frac{1}{3-0}+\frac{1}{3-1}+\frac{1}{3-2}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof by induction that $1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = \frac{n(2n-1)(2n+1))}{3}$ I need to know if I am doing this right. I have to prove that
$1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = \frac{n(2n-1)(2n+1))}{3}$
So first I did the base case which would be $1$.
$1^2 = (1(2(1)-1)(2(1)+1)) / 3
1 = 3/3
1 = 1$ Which is right... | Let your statment be $A(n)$. You want to show it holds for all $n \in \mathbb{N}$. You use the principle of induction to establish a chain of implications starting at $A(1)$ (you did that one).
What is left to show is
$$
A(n) \Rightarrow A(n+1)
$$
This means you consider $n$ fixed and try to proof $A(n+1)$. For this yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Is there a math function to find an element in a vector? I would like to write mathematically, if possible, the following statement:
Given a vector $x=[1,4,5,3]$ and an integer $j=3$, find the position of $j$ in $x$?
How to write this mathematically?
If I am looking for the position of the minimum value in $x$, I wo... | If you consider the vector $x$ as a function from $[1, n]$ to $\mathbb{N}$, you can use the inverse $x^{-1}$. See https://en.wikipedia.org/wiki/Inverse_function#Preimages.
In your example, if $x=(1,4,5,3)$ then $x^{-1} (\lbrace 3 \rbrace) = \lbrace 4 \rbrace$.
As Jonathan Gafar remarked, the inverse set might contain m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 10,
"answer_id": 4
} |
The textbook's way of deriving a natural deduction proof of $\vdash((\phi\leftrightarrow\psi)\leftrightarrow\phi)$ feels wrong. The problem is
"Show that if we have a derivation $D$ of $\psi$ with no undischarged assumptions, then we can use it to construct, for any statement $\phi$, a derivation of $((\phi\leftrightar... | There is no error; an allowed way to use $\to$-intro is to discharge a "non-existent" assumption, i.e.:
$\psi \vdash \phi \to \psi$,
as in the two top-right branches.
This corresponds to the fact that $\psi \to (\phi \to \psi)$ is a tautology; thus, assuming $\psi$, by modus ponens $(\phi \to \psi)$ follows, for $\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1719906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The "obvious" symmetry group $C_3 \times S_4$ related to the hexacode I am studying the large Mathieu groups and more specifically the hexacode from Robert Wilson's book "The Finite Simple Groups". The following paragraph is from page 184:The hexacode
My question is related to what is underlined: How should I interpret... | Write $a:=(12)(34)$, $b:=(135)(246)$, and $c:=(13)(24)$. Let $G=\langle a,b,c\rangle$. Note that $a^c=a$ while $(a^b)^c=a^{b^2}$. It follows that $N:=\{1,a,a^b,a^{b^2}\}$ is a normal subgroup of $G$ of order $4$. (Isomorphic to a Klein group.) Moreover, $b^c=b^{-1}$ so $H:=\langle b,c\rangle$ is isomorphic to $S_3$. M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1720006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that if $\sum_0^\infty f_n$ converges uniformly on $D$ to f and $f_n$ is bounded, then f is bounded. "Prove that if each $f_n$ is a bounded function and $\sum_0^\infty f_n$ converges uniformly on $D$ to $f$, then
$f$ is a bounded function"
I don't know how to do this at all. Any help appreciated.
| By Cauchy's Convergence Principle of uniformly convergence, $\forall\varepsilon>0\exists N>0$ s.t.$\forall n,m>N$,$|f_n-f_m|<\varepsilon,\forall x\in D$. In particular, $|f_n-f_{N+1}|<\varepsilon,\forall x\in D$. Thus $|f_n|<sup|f_{N+1}|+\varepsilon$ i.e. $f_n$ is uniformly bounded. Then you can use the definition of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1720105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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