Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is it possible to get the largest conceivable score in Hearts? I should explain the necessary rules: in the four-player game of Hearts, the object is to get as few points as possible. The points you receive are determined by the cards you pick up through a hand: hearts are worth $1$ point each, and the queen of spades ... | After each hand, the sum of all the scores must be a multiple of 26.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The Largest Gaps in the History of Mathematics Edit: Based on the useful comments below. I edited the original post in order to seek for other important historical gaps in mathematics.
Mathematics is full of the historical gaps.
The first type of these gaps belongs to the statements which has been proved long after th... | The Kepler conjecture:
No arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements.
The proof was overwhelmingly complex. Only recently a formal proof has been finished.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that the function 1/t is not in L2 (0,1] Need some help getting started with this problem:
$$f(t) = \frac{1}{t}$$
Show that $f(t)$ is not in $L_2(0,1]$, but that it is in the Hilbert space $L_{2}w(0,1)$ where the inner product is given by $$\langle x,y\rangle = \int(x(t)\overline{y(t)}w(t)dt$$ where $w(t)=t^2$.... | SOME HELP: $\int 1/t^2\,dt$ is divergent on $(0,1]$.
| {
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Separable Subset of the Real Line Consider $\mathbb{R}$ with the usual topology. Let $A$ be an infinite subset of $\mathbb{R}$. Prove that $A$ is separable.
So, if $A$ is to be separable, it must contain a countably dense subset $H$. I need to show that this $H$ exists to prove that $A$ is separable. If I don't know mo... | Since $\mathbb{R}$ is second countable, it has a countable basis. Intersecting each such basis element with $A$ is a basis of $A$ itself, so $A$ is also second countable. The standard proof applies: for each basis element $B$, choose some $x_B\in B\cap A$, and let $D$ be the set of all such $x_B$. This $D$ is counta... | {
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For a measure zero set $A$, the union $A\cup B$ has zero measure if and only if $B$ does Definition: A set $A$ has measure $0$ iff $\forall \epsilon > 0, \exists$ system of intervals $(I_\tau): A \subseteq \cup_\tau (I_\tau), 0 \leq \sum_\tau (\operatorname{length}(I_\tau)) < \epsilon$.
Using this definition, prove tha... | Yes, this proof works. The notes I would have towards it are:
It should be fairly obvious that, the total length in the union of two systems of intervals is exactly the sum of the total length in each interval - after all, those are both arising from absolutely convergent sums, so summing the results is the same as sum... | {
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How can this English sentence be translated into a logical expression? ( Translating " unless")
You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.
Let:
*
*$P$ stands for "you can ride the roller coaster"
*$Q$ stands for "you are under 4 feet tall"
*$R$ stands ... | The suggestion of $P\to (Q \wedge R)$ would say that in order to ride the roller coaster you must be at least $4$ feet tall and you must me at least $16$ years old. But I would say the meaning of the given sentence is that you need to satisfy one of the age and height conditions, not both.
I think the sentence means: ... | {
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Probability that a student guesses an answer (for multiple independent instances)? I read this question recently: Probability that a student knows the answer
Putting this in reverse, if the student guessed the answer then the probability would be:
\begin{eqnarray*}
A &=& \mbox{Student knows the correct answer} \\
C &=&... | Unfortunately, it's not that simple. You have to use Bayes' Theorem again. Firstly, since all answers are made independently of each other, given that the student is guessing, the probability of answering all $n$ questions correctly is $P(E \mid D^c) = \left(\frac{1}{4}\right)^n$. So,
\begin{eqnarray*}
P(D^c \mid E) &=... | {
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Can a collection of random processes be not random? A friend and I were having a debate about randomness and at one point, I said that it was possible to have a collection of random processes which were not random when "put together." He disagreed.
So, I put the question here more concretely and with more detail.
Su... | You are correct. The key is using dependence. For example, let $X$ be distributed as a continuous uniform random variable on the interval $[0,1].$ Let $Y=1-X.$
Then define $Z=X+Y.$ Now $Z$ is a constant, but composed of two random components.
There are more practical examples. Imagine a closed-loop system where compo... | {
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Is it possible to write a sum as an integral to solve it? I was wondering, for example,
Can:
$$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$
Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals.
But actually writing an integral form. Like
$$\displaystyle \sum_{n=1}^{\... | We can indeed write the sum as an integral, after research. Consider:
Find: $\psi(1/2)$
By definition:
$$\psi(z+1) = -\gamma + \sum_{n=1}^{\infty} \frac{z}{n(n+z)}$$
The required $z$ is $z = -\frac{1}{2}$
so let $z = -\frac{1}{2}$
$$\psi(1/2) = -\gamma + \sum_{n=1}^{\infty} \frac{-1}{2n(n - \frac{1}{2})}$$
Simplify th... | {
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Examples of mathematical discoveries which were kept as a secret There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret.
For example it is completely expected that if some mathematician find a proof of $P=NP$, he is not allowed by the government t... | In the 1920s Alfred Tarski found proofs that if a system of logic had either {CpCqp, CpCqCCpCqrr} or {CpCqp, CpCqCCpCqrCsr} as theses of the system, then it has a basis which consists of a single thesis. In other words, if both of {CpCqp, CpCqCCpCqrr} or both of {CpCqp, CpCqCCpCqrCsr} belong to some system S, then S h... | {
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Solving equations with mod So, I'm trying to solve the following equation using regular algebra, and I don't think I'm doing it right: $3x+5 = 1\pmod {11}$
I know the result is $x = 6$, but when I do regular algebra like the following, I do not get 6:
$3x=1 - 5\pmod{11}$
$x = \dfrac{(1 \pmod{11} - 5)} 3$
So, I figured... | The division in $\mathbb{F_{11}}$ is not like the division in $\mathbb{Q}$ !!! If you want to find $3(mod \; 11)^{-1}$ think of which element of $\mathbb{Z_{11}}$ multiplied by $3$ gives $1 \; (mod \; 11)$? It's $4$. So $$3x \equiv -4 \; (mod \; 11) \Leftrightarrow x \equiv -4*4 \; (mod \; 11)$$
And $-16 \equiv 6 \; (m... | {
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prove that a space is not connected Let $S=\{(x,0)\} \cup\{(x,1/x):x>0\}$. Prove that $S$ is not a connected space (the topology on $S$ is the subspace topology)
My thoughts: Now in the first set $x$ is any real number, and I can't see that this set in open in $S$. I can't find a suitable intersection anyhow.
| The set $\{(x,0) : x\in\mathbb R\}$ is open in $S$ because every point $(x,0)$ has an open neighborhood that does not intersect the graph of $y=1/x$. Just use $I\times\{0\}$ where $I$ is any open interval containing $x$.
Then do a similar thing with the set $\{(x,1/x): x>0\}$.
| {
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Slight confusion on Hom functor for a group seen as a category Let $(G,\cdot)$ be a group and let $BG$ be the category of this group with one formal object $*$ and the elements of $G$ as morphisms.
Now take the the covariant hom-functor $\text{Hom$(*,\_)$}:BG \to \mathbf{Set}$
$*\mapsto \text{Hom$(*,*)$} = [\text{the... | In general, if $X$ is an object and $g : Y \to Z$ is a morphism, then $\hom(X,Y) \to \hom(X,Z)$ is defined by $h \mapsto g h$. This doesn't change if $X$ is the only object. Hence, your map is left multiplication with $g$ (this is a bijection, but not a homomorphism unless $g=1$). Hence, the functor $\hom(\star,-)$ is ... | {
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Prove that: $\lim_{x\to 0}\frac{x}{\sin^2(x) + 1} = 0$ Prove
$$\displaystyle \lim_{x\to 0} \frac{x}{\sin^2(x) + 1} = 0$$
The proof:
Let $$|x| \le 1 \implies -1 \le x \le 1$$
$$\displaystyle \frac{|x|}{|\sin^2(x) + 1|} < \epsilon\text{ for }\displaystyle |x| < \delta$$
$$-1 \le x \le 1
\\\implies \sin(-1) \le \sin(x) \... | This is way too complicated, don't you think?
Why not just say that
$$
\left| \frac{x}{1+\sin^2 x}
\right| \le |x| \le \epsilon
$$
as
soon as $|x|<\delta = \epsilon$?
| {
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Does an uncountable Golomb ruler exist? Does there exist an uncountable set $G\subset \mathbb{R}$ such that, for $a,b,c,d \in G$, if $a-b=c-d$ then $a=c$ and $b=d$?
| I think so (assuming Choice). Let $G$ be a vector space basis of $\Bbb{R}$ over $\Bbb{Q}$. Then $G$ is, among other things, linearly independent over $\Bbb{Z}$ and uncountable.
| {
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Eigenspace of finite abelian group Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$.
Is it true that $V$ is also eigenspace for all $G$ (that is $\rho(g)v=\lambda_g v$ for all $v\in V$ and $g\in G$)?
| No.
For a counterexample, take any faithful representation of a nontrivial group (simplest is an action of $\Bbb Z_2$ on $\Bbb C^2$, say, by reflection through a line), and consider the unit element, that has the whole space as eigenspace, while other elements can have more complex eigenspace decomposition.
| {
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Give a partition of ω I have a question which I'm deeply confused about. I was trying to do some problems my professor gave us so we could practice for exam, one of them says:
Give a partition of ω in ω parts, everyone of them of cardinal ω.
I know that $ ω=\left \{ 1,2,3,...,n,n+1,.... \right \}$ , but I thought t... | There are many ways to do it, since $|\omega\times\omega|=\omega$. Here’s one. Every $n\in\Bbb Z^+$ can be written uniquely in the form $n=2^km$, where $m$ is odd, and you can start by letting $S_k=\{2^km:m\in\omega\text{ is odd}\}$: $S_0$ is the set of odd positive integers, $S_1$ the set of even positive integers tha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of divergence/convergence of a series Consider the series :
$$\sum\limits_{n = 1}^\infty {\frac{{(n + 2)^3 n^\alpha }}{{\sqrt[3]{{n^2 + 4n + 7\,}}\sqrt {n + 1} }}} $$
where $
\alpha \in \Re $ . I managed to determine that when $
\alpha \ge \frac{{ - 13}}{6}$ the series diverges, but what about the other case... | Hint : Divide denominator and numerator by $n^{(\frac{2}{3}+\frac{1}{2})}=n^\frac{7}{6}$
| {
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Limit with arctan: $\lim_{x\rightarrow 0} \frac{x\sin 3x}{\arctan x^2}$ $$\lim_{x\rightarrow 0} \frac{x\sin 3x}{\arctan x^2}$$
NB! I haven't learnt about L'Hôpital's rule yet, so I'm still solving limits using common limits.
What I've done so far
$$\lim_{x\rightarrow0}\left[\frac{x}{\arctan x^2}\cdot 3x\cdot \frac{\sin... | You should put in
$$
\frac{x^2}{\arctan x^2}
$$
that has limit $1$:
$$
\lim_{x\to0}\frac{x\sin3x}{\arctan x^2}=
\lim_{x\to0}3\frac{\sin3x}{3x}\frac{x^2}{\arctan x^2}=\dots
$$
If you don't know the limit above, just substitute $t=\arctan x^2$, so $x^2=\tan t$ and the limit is
$$
\lim_{x\to0}\frac{x^2}{\arctan x^2}=\lim_... | {
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Find $\int \ln(\tan(x))/(\sin(x) \cos(x))dx$ I was given this question in a review package, and it has me stumped:
I started off using the identity $\tan(x) = \sin(x) / \cos(x)$ and then used the fact that $\sin(x) \cos(x) = .5\sin(2x)$ to try and simplify the denominator. I looked around for a basic $u$ substitution ... | \begin{eqnarray}
\int\frac{\ln\tan x}{\sin x\cos x}dx&=&\int\frac{\ln\tan x}{\tan x}\sec^2xdx
=\int\frac{\ln\tan x}{\tan x}d\tan x\\
&=&\int\ln\tan xd\ln\tan x=\frac{1}{2}(\ln\tan x)^2+C
\end{eqnarray}
| {
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"question_score": "6",
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"answer_id": 2
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Solve $y'' - y' = yy'$ and find three Other Distinct Solutions Been stuck on this for a while. I need to solve the following differential equation by finding the constant solution y = c and three other distinct solutions.
$$y'' - y' = yy'$$
If someone could give me a complete step by step explanation, it would be great... | I'll assume the independent variable is $x$, you can make the necessary changes if it's $t$ or anything else.
Let $z=y'$. Then
$$y''=\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}=z\frac{dz}{dy}$$
and substituting into the DE gives
$$z\frac{dz}{dy}-z=yz\ .$$
See if you can take it from here.
| {
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What is the order of the alternating group $A_4$? When I write out all the elements of $S_4$, I count only 11 transpositions. But in my text, the order of $A_4$ is $12$. What am I missing?
$A_4=\{(12)(34),(13)(24),(14)(23),(123),(124),(132),(134),(142),(143),(234),(243)$
$|A_4|=11$
| The order of $A_n$ is always half the order of $S_n$, consider the bijective map from the even permutations to the odd permutations where $\varphi(\pi)=(12)\pi $. This is a bijection since the inverse is the map from the odd permutations to the even permutations $\varphi^{-1}(\pi)=(12)\pi$.
| {
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Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$ Show that
$$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$
How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm.
| Hint: Use that
$$\log(1 + e^x) = \log[e^x (1 + e^{-x})]$$
$$\cdots= \log e^x + \log(1 + e^{-x}) = x + \log(1 + e^{-x}).$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine "winner" in exponential contest If I have one light bulb that could be one of $2$ kinds, ($A$ and $B$ are the lifetimes of first and second type: $A\sim \exp(1)$ and $B\sim \exp(3)$), and each time a bulb dies, another bulb replaces it (with probability $0.5$ to be $A$ or $B$).
$X$ is the lifetime of the ligh... | You have $\pi(A)=\pi(B)=0.5$. You want the posterior value $$\pi(A\mid\text{failure at }t)=\dfrac{\pi(A)f_A(t)}{\pi(A)f_A(t)+\pi(B)f_B(t)}.$$
The densities are $f_A=e^{-t}$ and $f_B=3e^{-3t}$ so $$\Pr(\text{type }A\mid\text{failure at }t) = \dfrac{0.5 \times e^{-t}}{0.5 \times e^{-t}+0.5 \times 3e^{-3t}}= \dfrac{1}{1+... | {
"language": "en",
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Symmetric matrices with the eigenvalues comparable Let $A,B$ be $n\times n$ real symmetric matrices, with eigenvalues $\lambda_i$ and $\mu_i$ respectively, $i=1,\cdots,n$. Suppose that
$$\lambda_i\leq\mu_i,\forall\ i.$$
Show that there exists an orthogonal matrix $O$ such that
$$O^TBO-A$$
is non-negative definite.
I ... | Let $P,Q$ be orthogonal with $P^tAP$ the diagonal matrix with diagonal $\lambda_1,\dots,\lambda_n$ and $Q^tBQ$ the diagonal matrix with diagonal $\mu_1,\dots,\mu_n$. Then $Q^tBQ-P^tAP$ is non-negative definite, so $PQ^tBQP^t-A$ is. Let $O=QP^t$.
| {
"language": "en",
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Finding a bound for $\sum_{n=k}^l \frac{z^n}{n}$ For $z\in\mathbb{C}$ such that $|z|=1$ but $z\neq1$ and $0<k<l$, I'm trying to prove that:
$$\left|\sum_{n=k}^l \frac{z^n}{n}\right| \leq \frac{4}{k|1-z|}$$
It's more of a game that slowly frustrates me... I've got
$$\left|\sum_{n=k}^l \frac{z^n}{n}\right|=\left|\frac{... | Start from
$$|1-z|\cdot\left|\sum_{n=k}^l \frac{z^n}{n}\right|=\left|\sum_{n=k}^l\frac{z^n}n-\sum_{n=k+1}^{l+1}\frac{z^n}{n-1}\right|.$$
The term for $n=k$ and $n=l+1$ can be bounded by $1/k$. To conclude, notice that
$$\left|\sum_{n=k+1}^lz^n\left(\frac 1n-\frac 1{n-1}\right)\right|\leqslant \sum_{n=k+1}^l\left|z^n\... | {
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How can I measure a frustum inside a frustum? If I know the measurements of a frustum A, how can I find the measurements of frustum B if I only know B's bottom radius, slanted side angle and volume?
This problem arose after finding how deep the booze in my cocktail glass is, but when not filling the glass to the brim.
... | So when you are finding the area of a frustum, you are basically finding the volume of one cone and subtracting it from the volume of another cone. So, when doing this problem, just find the area of the cone formed at a given height h and subtract it from the cone at the bottom with a radius of 40.
| {
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How would chemical equations be balanced with matrices? For example, I have this equation:
$$\mathrm{KMnO_4 + HCl = KCl + MnCl_2 + H_2O + Cl_2}$$
Then I get this:
$$a \cdot \mathrm{KMnO_4} + b \cdot \mathrm{HCl} = c \cdot \mathrm{KCl} + d \cdot \mathrm{MnCl_2} + e \cdot \mathrm{H_2O} + f \cdot \mathrm{Cl_2}$$
$$
\beg... |
K: a = c Mn: a = d O: 4a = e H: b = 2e Cl: b = c + 2d + 2f
How would I get the values of a, b, c, d, e, and f from here?
Well... Reading the equations in the order they were given and using a as a parameter, one gets successively c = a, d = a, e = 4a, b = 2e = 8a, and 2f = b - c - 2d = 8a - a - 2a = 5a.
This is solve... | {
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Smoothness of Fourier transform of a measure Is the Fourier transform of a finite Borel measure on $\mathbb{R}$ necessarily a smooth function?( $\widehat{\mu}(x)=\int_\mathbb{R}e^{-i\pi xy} d\mu(y)$)
| No. It's continuous, but in general not smooth.
If we take for $\mu$ the measure given by the density
$$f(x) = \frac{1}{1+x^2}$$
with respect to the Lebesgue measure, we find that
$$\hat{\mu}(y) = \pi e^{-\lvert y\rvert},$$
which is not differentiable at $0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Can basis vectors have fractions? So I was diagonalizing a matrix in a book, and one of the basis vectors was [3/2, 1], after doing the problem, the answer in the book was different than mine. It came with an explanation, and in it the basis vector was [3,2]. They are the same thing, just multiples of each other, so I ... | You are right and they are also right. This is because multiplication by non-zero constants does not affect the span of the basis nor the linear independence of the vectors in that basis. You should be able to see this quite obviously once you think about it.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Ball and urn problem Let's say there are three types of balls, labeled $A$, $B$, and $C$, in an urn filled with infinite balls. The probability of drawing $A$ is $0.1586$, the probability of drawing $B$ is $0.81859$, and the probability of drawing $C$ is $0.02275$.
If you draw $6$ balls, what's the probability that you... | Any sequence of the sort $ABACAB$ has the same probability to occur. So it comes to finding how many distinct words there are having $3$ times an $A$, $2$ times a $B$ and $1$ time a $C$.
You could start with $6$ open spots and then placing the $A$'s. That gives $\binom{6}{3}$ possibilities.
Then place the $B$'s. That... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1006209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Independence and uncorrelatedness between two normal random vectors. If $X$ and $Y$ are normal random vectors in $\mathbb R^n$ and in $\mathbb R^m$, and they are jointly normally distributed i.e. $(X,Y)$ is normally distributed in $\mathbb R^{n+m}$, then are the following equivalent
*
*$\operatorname{Cov}(X,Y)=0$;
... | $\newcommand{\E}{\operatorname{E}}\newcommand{\var}{\operatorname{var}}\newcommand{\cov}{\operatorname{cov}}$
In comments you say you have a theorem that if two multivariate normal distributions have the same mean and the same variance, then they are the same distribution.
You have
$$
\E\begin{bmatrix} X \\ Y \end{bmat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of passing this multiple choice exam
*
*A multiple choice exam has 175 questions.
*Each question has 4 possible answers.
*Only 1 answer out of the 4 possible answers is correct.
*The pass rate for the exam is 70% (123 questions must be answered correctly).
*We know for a fact that 100 questions we... | Hint : Use the formula for binomial distributed random variables.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with finding the $\lim_{x\to 0} \tan x \csc (2x)$ So I am trying to figure out the limit
$$\lim_{x\to 0} \tan x \csc (2x)$$
I am not sure what action needs to be done to solve this and would appreciate any help to solving this.
| Note that $\csc (2x) = \frac{1}{\sin(2x)} = \frac{1}{2 \sin x \cos x}$, and $\tan x = \frac{\sin x}{\cos x}$.
So $$\lim_{x \to 0} \tan x \csc (2x) = \lim_{x \to 0} \frac{1}{2 \sin x \cos x} \frac{\sin x}{\cos x} = \lim_{x \to 0} \frac{1}{2 \cos^2 x} = \frac{1}{2}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Fundamental theorem of finitely generated abelian groups. If $G :=\langle x,y,z \ | \ 2x+3y+5z = 0\rangle$ then find what group $G$ is isomorphic to.
I think I'm supposed to use the fundamental theorem of finitely generated abelian groups, but I don't know if I am using it correctly below.
Since $2x = 0$ we know that $... | This is not a rigorous answer, but it provides some intuition.
Per my comment, we can get $(1,1,-1)$ as a generator of a subgroup isomorphic to $\mathbb{Z}$.
We can also get $(4,-1,-1)$ as another generator of a subgroup isomorphic to $\mathbb{Z}$, and it should be clear that these two generators are independent (i.e. ... | {
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Inconsistency in two-sided hypothesis testing Suppose you have two sets of data with known population variances and want to test the null hypothesis that two means are equal, ie. $H_{0}: \mu_{1} = \mu_{2}$ against $H_{1}: \mu_{1} > \mu_{2}$. There's a certain way I want to think about it, which is the following:
\begin... | If is probably that $\mu_1<\mu_2$, then $\mu_1\geq\mu_2$ is rejected, then not is possible that holds $\mu_1<\mu_2$ and $\mu_1>\mu_2$ simultaneusly.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show the diffusion equation is a normalised distribution. The diffusion equation is defined to be $$P(x,t) = \dfrac{1}{\sqrt{4D\pi t}} \exp \left(-\dfrac{x^2}{4Dt}\right),$$ where $D$ is a physical constant.
Show that the reaction diffusion equation is a normalised distribution.
I take that this means that I need to sh... | $$
u(x,t) = \frac{1}{\sqrt{4D \pi t}}e^{-x^2/(4Dt)}
$$
is the fundamental solution to the initial value problem,
$$
\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2},\quad u(x,0) = \delta (x)\,.
$$
Note the following:
*
*For $\alpha>0$, it is well-known that
$$
\int\limits_{-\infty}^{\infty}e^{-... | {
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Maximization of a ratio Edit: Removed solved in title, because I realize I need someone to check my work.
Ok, so the problem is a lot more straight forward than I originally approached it (which was a false statement -- so it was excluded).
Question:
Let R,S, x $\in$ N with x $\le$ R*S and $0 \lt$ R $\le$ S. Next, def... | Consider the expression $\frac xB$. For B to be a multiplicative factor of x - c, c must equal the remainder between B and x. Therefore, we can rewrite c as c = x - dB, where d is the unique natural number satisfying both dB $\le$ x and (d + 1)B $\gt$ x. Substituting,
A = $\frac{x - (x - dB)}{B}$ = d.
However, d depen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Book recommend for topics of Integrals in multivariable calculus. I am an average student and have to study following topics on my own for the exam :
The measure of a bounded interval in $\mathbb R^n$ , the Riemann integral of a bounded function defined
on a compact interval in $\mathbb R^n$ , Sets of measure zero... | I would say Mathematical Analysis II by Zorich fits the bill.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Differentiate $f(x)=\int_x^{10}e^{-xy^2}dy$ with respect to $x$ I am trying to find $f'(x)$ when $0\leq x\leq 10$. I know I could use the formula given on this wikipedia page: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign but I have been asked to justify all steps of the calculation so this isn't... | There are indeed formula for the differentiation of functions of the form $x\mapsto \int_{a(x)}^{b(x)}f(x,t)\mathrm{dt}$ under some good conditions on the functions $a$, $b$ and $f$.
However, in our case, the function $f$ has a quite nice form. We can start from the substitution $xy^2=t^2$, hence $t=\sqrt x\cdot y$ wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1007206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$ Prove that:
$$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$
But I obtain this equal zero:
$$\frac {2\Gamma'(2z)}{\Gamma(2z)} - \frac {\Gamma'(z)}{\Gamma(z)} - \frac {\Gamma'(z+... | Since
$$\Gamma(z)\cdot \Gamma(z+1/2)=2\sqrt{\pi}\cdot 4^z\cdot \Gamma(2z) $$
by considering the logarithmic derivative of both sides we get:
$$\frac{\Gamma'}{\Gamma}(z)+\frac{\Gamma'}{\Gamma}(z+1/2)=2\frac{\Gamma'}{\Gamma}(2z)+2\log 2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1007258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality proof by induction, what to do next in the step I have to prove that for $n = 1, 2...$ it holds: $2\sqrt{n+1} - 2 < 1 + \frac{1}{\sqrt2} + \frac{1}{\sqrt3} + ... + \frac{1}{\sqrt{n}}$
Base: For $n = 1$ holds, because $2\sqrt{2}-2 < 1$
Step: assume holds for $n_0$.
$2\sqrt{n+2} - 2 < 1 + \frac{1}{\sqrt2} + \f... | For induction step, it's enough to prove $\frac{1}{\sqrt{n+1}}>2(\sqrt{n+2}-\sqrt{n+1})$.
$$2(\sqrt{n+2}-\sqrt{n+1})=\frac{2}{\sqrt{n+2}+\sqrt{n+1}}<\frac{2}{2\sqrt{n+1}}=\frac{1}{\sqrt{n+1}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1007371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the positive integer numbers to get $\frac{\pi ^2}{9}$ As we know, there are many formulas of $\pi$ , one of them $$\frac{\pi ^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}......
$$
and this $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}......$$
Now,find the positive integer numbers $(a_{0}, a_{... | Hint: $\frac 8 9 = \frac 1 2 + \frac 1 3 + \frac 1 {18}$ Now try using your second formula.
| {
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Trigonometric equation, missing some solutions I'm missing part of the answer, and I'm not quite sure why. The given answer doesn't even seem to hold...
Solve for x: $$\tan 2x = 3 \tan x $$
First some simplifications:
$$\tan 2x = 3 \tan x $$
$$\tan 2x - 3 \tan x = 0$$
$$\frac{\sin 2x}{\cos 2x} - \frac{3 \sin x}{\cos ... | Setting $\tan x=t$
we have $$\frac{2t}{1-t^2}=3t\iff2t=3t(1-t^2)\iff t(2-3+3t^2)=0$$
If $t=0,\tan x=0, x=n\pi$ where $n$ is any integer
$2-3+3t^2=0\iff 3t^2=1\implies\cos2x=\dfrac{1-t^2}{1+t^2}=\dfrac12=\cos\dfrac\pi3$
$\implies2x=2m\pi\pm\dfrac\pi3$ where $m$ is any integer
| {
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Expected number of rolls A fair m-sided dice is rolled and summed until the sum is at least N. What is the expected number of rolls? In other words what is the number of rolls if we roll a m-sided dice and the sum of rolls become at least N.
| If $f(N)$ is the expected number of rolls, by conditioning on the first roll we have
$f(N) = 1 + m^{-1} \sum_{j=1}^m f(N-j)$ for $N > 0$, with $f(N) = 0$ for $N \le 0$. The generating function is $$g(z) = \sum_n f(n) z^n = \dfrac{mz}{m - (m+1)z + z^{m+1}}$$
EDIT: If you're interested in the asymptotic behaviour of $f... | {
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Equality of exponential functions from geometric series I'm currently trying to understand why the first and second line of this equation
are in fact equal. This is taken from "Introduction to the Physics of Waves" by Tim Freegarde from a chapter about diffraction gratings. The notation is somewhat ambiguos (I think)... | Hint: Multiply the numerator and denominator by the factor $-e^{ikd \sin(\vartheta)/2}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Study of a function on interval $[0,1]$ Let $f(x)$ be a function defined on the interval $[0,1]$ such that
$$x \mapsto \dfrac{x^2}{2-x^2}$$
Show that, for all $x \in [0,1[,\ 0\leq f(x)\leq x <1.$
Attempt:
Let $$g(x)=\dfrac{f(x)}{x} $$
$$g'(x)=\dfrac{2+x^2}{(2-x^{2})^2}\geq 0 $$
i'm stuck here
Thanks for your help
| $$0\leqslant x<1\Rightarrow 0\leqslant x^2<x<1\\2-x^2>1\\\frac{1}{2-x^2}<1\\\text{multiply by }x^2\\x^2\frac{1}{2-x^2}<x^2*1\\\frac{x^2}{2-x^2}<x^2\leqslant x <1\\$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding zeroes of a complex function over a lattice Question:
Let $L = \mu\mathbb{Z}[i]$ be a lattice in $\mathbb{C}$, where $\mathbb{Z}[i] = \{n+mi:n,m\in\mathbb{Z}\}$ and $\mu \in \mathbb{R}_{+}$.
Let $$\mathrm{G}_k = \displaystyle\sum_{\omega \in L}_{\omega \not= 0} \dfrac{1}{\omega^k} $$
be the corresponding Eisen... | For an explicit formula of the zeroes of the Weierstrass $\wp$-function $P_L(z)$ see the article of Eichler and Zagier here. We have $\tau=i$ for $\mathbb{Z}[i]$.
Theorem: The zeroes of $P_L(z)$ are given by
$$
z=m+\frac{1}{2}+ni\pm \Biggl(\frac{\log(5+2\sqrt{6})}{2\pi i}+144\pi i\sqrt{6} \int_i^{i\infty}(t-i)\frac{\D... | {
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Help proving $\partial (A \cup B) = \partial A\cup\partial B$? I know this is a duplicate but the other two haven't helped me much.
Fist attempt: Tried proving through double inclusion, but wasn't sure of how to convey being an element of one implied being an element of the other in either direction, although I suspe... | It is false.
Consier $\Bbb R$ with usual topology, $A=[0,2]$, $B=[1,3]$.
$$\partial(A\cup B)=\{0,3\}$$
$$\partial A\cup\partial B=\{0,1,2,3\}$$
Just think that some of the border of $A$ can be in the interior of $B$.
| {
"language": "en",
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An annoying Pell-like equation related to a binary quadratic form problem Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $.
Show by elementary means that the Diophantine equation
$$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$
has an integer solution (that is, a solution $(x,y)\in\mathbb Z^2$). If po... | You have the binary quadratic form
$$ \color{red}{ f(x,y) = B x^2 + (D-A)xy - C y^2} $$ in your last paragraph. The discriminant is $$ \Delta = (D-A)^2 + 4 B C. $$
You also have $AD-BC = 1$ and $A+D = -1.$ So, $BC - AD = -1$ and
$$ A^2 + 2 AD + D^2 = 1, $$
$$ 4BC - 4AD = -4, $$
$$ A^2 - 2 AD + D^2 +4BC = -3, $$
$$... | {
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The group $\mathbb{Z}_{10}$ has precisely $4$ subgroups True/False: The group $\mathbb{Z}_{10}$ has precisely $4$ subgroups.
Solution:
True, since there are $4$ divisors $1,2,5,10$ thus it has $4$ subgroups.
| You are correct.
If you define $\mathbb{Z}_{10} = \mathbb{Z}/ 10\mathbb{Z}$, then this follows from the correspondence theorem: the subgroups of $\mathbb{Z}_{10}$ correspond to the subgroups of $\mathbb{Z}$ that contain $10\mathbb{Z}$, and there is exactly one such subgroup for each divisor of $10$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Linear independence of $\sin^2(x)$ and $\cos^2(x)$ The Wronskian for $\sin^2x, \cos^2x$ is
\begin{align}
& \left| \begin{array}{cc} \sin^2 x & \cos^2 x \\ 2\sin x\cos x & -2\cos x\sin x \end{array} \right| \\[8pt]
= {} & -2\sin^2x \cos x \sin x - 2 \cos^2 x \sin x \cos x,
\end{align}
with $x = \frac{π}{6},$ this is $=$... | It suffices to show that the Wronskian is not zero for a single value of $x$. We have: $$W(x) = \begin{vmatrix} \sin^2x & \cos^2x \\ 2\sin x \cos x & -2 \sin x \cos x\end{vmatrix} = -2\sin^3x \cos x - 2\sin x \cos^3 x$$
$$W(x) = -2\sin x \cos x = -\sin(2x)$$
Then, $W(\pi/4) = -1 \neq 0$, so the functions are linearly i... | {
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"source": "stackexchange",
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Humorous integration example? I was just reading though an introductory calculus book and it has the note:
NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in integration than it is in differentiation.
Now that's fair enough to point out and it gives a ... | It is easy to verify that, in the version suggested by @MPW, for $f=e^{ax}$, $g=e^{bx}$, we have
$$
a=\frac{b}{b-1}.
$$
For $b=2$ we have the answer obtained by Sam.
Edit:
There is a sketch, that this is near general solution.
We denote integral by $F$, as it is up to constant some function. Then we have
$$
F(fg)=F(f)... | {
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"timestamp": "2023-03-29T00:00:00",
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Complex analysis, showing a function is constant Let $\Omega$ be the right half plane excluding the imgainary axis and $f\in H(\Omega)$ such that $|f(z)|<1$ for all $z\in\Omega$. If there exists $\alpha\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $$\lim_{r\rightarrow\infty}\frac{\log|f(re^{i\alpha})|}{r}=-\infty$$ prov... | I can only give a partial proof.
Let $g_n(z)=f(z)e^{nz}$, $n=1,2,\ldots$, and $g_n(z)\in H(\Pi)$.
Assume $g_n(z)\in C(\bar\Pi)$.
Let $K_1=\{re^{i\theta}\mid\theta\in(\alpha,\pi/2)\}$, $K_2=\{re^{i\theta}\mid-\theta\in(\alpha,\pi/2)\}$ and $K_3=\Pi\setminus(K_1\cup K_2)$. By the assumption $\lim_{r\rightarrow\infty... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that no hamiltonian cycle exists in the graph ** Show that the graph below has a hamiltonian path but no hamiltonian cycle.
You can find more than one hamiltonian path such as $(b,a,c,f,e,g,d)$.
Actually, I tried many times to find a hamiltonian cycle, but I couldn't find a cycle. The problem is how to pr... | Notice that if you delete the edge joining $g$ and $f$, then you get the complete bipartite graph $K_{3,4}$ with sides $\{g,a,f\}$ and $\{b,d,c,e\}$.
It may help now to redraw the graph in the usual way $K_{3,4}$ is drawn. In any case, now you need prove that there is no Hamilton cycle in $K_{3,4}$ and no Hamilton path... | {
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"timestamp": "2023-03-29T00:00:00",
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what is the sum of square of all elements in $U(n)$? I know that $\sum\limits_{a\in U(n)} a=\frac{n\varphi(n)}{2}$ where $U(n):=\{1\leq r\leq n: (r, n)=1\}$ is a multiplicative group. And I know how to prove this result.
What I was willing to know was this $\sum\limits_{a\in U(n)} a^2$. is it possible to find in closed... | See OEIS sequence A053818 and references there.
| {
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How Find the value $\det{(A)}$ if know $A\cdot\begin{bmatrix}1 \end{bmatrix}$ let $A_{n\times n}$matrix,and $A^{*}$is Adjugate matrix of the $A$,$p,q>0$ is give numbers,and such following condition
$$A\cdot\begin{bmatrix}
1\\
1\\
\vdots\\
1
\end{bmatrix}=\begin{bmatrix}
p\\
p\\
\vdots\\
p
\end{bmatrix},and ,A^{*}\cdot\... | There is a theorem which states that
$$A \cdot A^*= A^* \cdot A= \det(A) \cdot I$$
Where $I$ is the identity matrix.
So, applying $A \cdot A^*$ to the vector $[1, \dots, 1]^T$ you get that the determinant of $A$ is $pq$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1008648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that $\frac{\pi ^2}{p}\neq \sum_{n=1}^{\infty }\frac{1}{a_{n}^2}$ we have many formula for $\pi ^2$ as follow $$\frac{\pi ^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}....$$
and $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}....$$ and for $\frac{\pi ^2}{9}$, $\frac{\pi ^2}{12}$, $\frac{\pi ^2}{... | I ran the greedy algorithm from the previous question for the first 1000 primes with the added caveat that the terms in the sequence have to be distinct. It appears that your statement is only true for the primes 2, 3, 5, 11, 13 and false for the rest! I've only used 40 terms in the algorithm for each of the primes. Fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Examples of the Geometric Realization of a Semi-Simplicial Complex I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions:
I find it difficult to visualize without specific examples. Can anyone help to provide some typical examples of geometric realization of semi-... | Let $K_0=\{a,b\}$ and let $K_1=\{X,Y\}$ with $\partial_0(X)=\partial_1(Y)=a$ and $\partial_1(X)=\partial_0(Y)=b$. This is a $\Delta$-complex.
The space $\bar{K}$ is a disjoint union of two discrete points given by $a$ and $b$, and two disjoint intervals given by $X$ and $Y$. The equivalence relation just says that we j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1008871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to integrate the dilogarithms? $\def\Li{\operatorname{Li}}$
How can you integrate $\Li_2$? I tried from $0 \to 1$
$\displaystyle \int_{0}^{1} \Li_2(z) \,dz = \sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$
$$\frac{An + B}{n^2} + \frac{D}{n+1} = \frac{1}{n^2(n+1)}$$
$$(An + B)(n+1) + D(n^2) = 1$$
Let $n = -1, \implies D = 1... | Maybe you should look at your decomposition as
$$\frac1{n^2 (n+1)} = \frac1{n^2} - \frac1{n (n+1)}$$
The sum over the second term is easy, given that the indefinite sum is telescoping, i.e.,
$$\sum_{n=1}^N \frac1{n (n+1)} = 1-\frac1{N+1}$$
We take the limit as $N \to \infty$ and then we may view this as the infinite s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$L^{2}$ convergent, subsequence, directed family of points I have a question about a convergence.
Let $(E,\mathcal{B},m)$ be a measure space. I think the following assertion is very famous:
Let $f_{n},f \in L^{2}(E;m)\quad(n=1,2,\cdots)$. If $f_{n}\to f $ in $L^{2}(E;m)$ then there exists subsequence $(f_{n_{k}})_{k=1... | No. All you can conclude is that for every sequence $\{t_n\}$, $t_n\to0$, there is a subsequence $\{t_{n_k}\}$ such that $f_{t_{n_k}}\to f$ $m$-a.e.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1009029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Analogy to the purpose of Taylor series I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and this is my attempt to re-learn Calculus in a better way, to understand Physics a... | I can't think of any good analogies.
Perhaps the best way to get comfortable with Taylor Series is to look at some interactive examples:
http://demonstrations.wolfram.com/GraphsOfTaylorPolynomials/
Starting with $f(x) = e^x$, step up the highest number of terms from 1 to 10 to see how higher polynomials do an increasi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How to find the determinant of this $(2n+2)$ x $(2n+2)$ matrix? I need to calculate the determinant of the following matrix:$$\begin{bmatrix}0&0&-2x_1& \cdots &-2x_n&0& \cdots &0\\0&0&0& \cdots&0&-2x_1& \cdots&-2x_n\\-2x_1&0&-1& \cdots&0&1& \cdots&0\\ \vdots&\vdots &\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\-2x_n&0&0&... | Write w.l.o.g. your matrix in the form
$$
A:=\begin{bmatrix}
0 & 0 & x^T & 0 \\
0 & 0 & 0 & x^T \\
x & 0 & -I & I \\
0 & x & I & -I
\end{bmatrix}.
$$
Take any nonzero vector $z$ orthogonal to $x$ ($z^Tx=0$), there is a whole $(n-1)$-dimensional subspace of them. Now
$$
A\begin{bmatrix}
0 \\ 0 \\ z \\ z
\end{bmatrix}... | {
"language": "en",
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What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)? What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)?
Further, what axioms (or properties) of $\mathbb R $ do these topological properties depend on? (I suppo... | One topological property of $\mathbb{R}$ that makes it uncountable (more precisely, of size at least contiuum) is connectedness. Any Tychonoff (or even functionally Hausdorff) connected at least two point space has size at least continuum. The connectedness comes from completeness, actually the order completeness and n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Median of truncated / limited normal distribution The peoples weight is normally distributed $\mathcal{N}(0,\,1)$
The $\mu \; , \; \sigma \; and \; \sigma^2$ are known.
How can i calculate the median weight of people if everyone who weights less than amount of x is removed / ignored.
I would appreciate some hints on ... | Let $F$ denote the CDF of the uncensored weight, then the median $m_x$ of the weight censored below the value $x$ solves $F(m_x)=\frac12(1+F(x))$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is irrational times rational always irrational? Is an irrational number times a rational number always irrational?
If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number?
| If $a$ is irrational and $b\ne0$ is rational, then $a\,b$ is irrational. Proof: if $a\,b$ were equal to a rational $r$, then we would have $a=r/b$ rational.
| {
"language": "en",
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"source": "stackexchange",
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$\int \frac{2^{\sin \left(\sqrt{x}\right)} \cos \left(\sqrt{x}\right)}{\sqrt{x}} \, dx$ I have been asked to integrate:
$$\int \frac{2^{\sin \left(\sqrt{x}\right)} \cos \left(\sqrt{x}\right)}{\sqrt{x}} \, dx$$
In such a small integration you dont have to write it down but to see where I am struggling I have provided a ... | $$I=\int\frac{2^{\sin\sqrt{x}}\cos\sqrt{x}}{\sqrt{x}}dx$$
$u=\sin\sqrt{x}\Rightarrow \frac{du}{dx}=\frac1{2\sqrt{x}}\cos\sqrt{x}$ and so: $dx=\frac{2\sqrt{x}}{\cos\sqrt{x}}du$ which converts our integral into:
$$I=2\int2^{u}du$$
now to integrate this notice that:
$$2^{u}=e^{\ln(2^{u})}=e^{u\ln(2)}$$
now if we make the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Could the Riemann hypothesis be provably unprovable? I don't know much about foundations and logic, so I ask forgiveness if my question is just plain stupid.
Assume we have a statement of the form:
There exist no $x\in X$ such that $P(x)$.
where $X$ is some kind of set (or class, or similar stuff) and $P$ is a set of... | You can check out the answers to this related MO question:
"Can the Riemann hypothesis be undecidable?"
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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how to find out if something in general(scalar, function, vector) is in the span of a set im pretty sure that if a vector is in the span of a set of vectors, then it can be written as a linear combination of the vectors in the set, whch you can find out by setting up a system of equations. But what if you have an arbit... | You do it the same way as with Euclidean vectors -- you check if you can represent a given function (/vector) by a linear combination of your basis:
That is, you try to solve:
$a)\ 1=a\cos^2(x)+b\sin^2(x)+c\tan^2(x)$
$b)\ \sec^2(x)=a\cos^2(x)+b\sin^2(x)+c\tan^2(x)$
$c)\ \cos(2x)=a\cos^2(x)+b\sin^2(x)+c\tan^2(x)$
$d)\ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What are the theorems in mathematics which can be proved using completely different ideas? I would like to know about theorems which can give different proofs using completely different techniques.
Motivation:
When I read from the book Proof from the Book, I saw there were many many proof for the same theorem usi... | A large rectangle is tiled with smaller rectangles. Each of the smaller rectangles has at least one integer side. Must the large rectangle have at least one integer side?
What if you replace 'integer' with 'rational' or 'algebraic'?
Here are fourteen proofs.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divergent Sequence for Wau So I just "learned" about the number Wau from Vi Hart's video. It's amusing, to be sure, but the actual "definition" she presents got me thinking.
We can formalize the construction in this way: set $x_0=\frac2{1+3}=\frac12$ and then
$$x_{n+1} = \frac{2}{\frac12x_n+\frac32x_n} = \frac{2}{2x_n}... | It may not be a correct intuition, but my idea is to perturb the iteration as follows: let $\epsilon > 0$ and define
$$ x_{0} = \tfrac{1}{2}, \qquad x_{n+1} = \frac{1}{x_{n}} + \epsilon. \tag{1}$$
Then it follows that $x_{n}$ converges to $\sqrt{1+\epsilon^{2}} + \epsilon$, which indeed converges to $1$ as $\epsilon \t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the geometric interpretation of general solution Here are the linear equation:
$$2x+z=0$$
$$-x+3y+z=0$$
$$-x+y+z=0$$
I have found that the general solution is,
$$t
\begin{bmatrix}
\frac{-1}2\\
\frac{-1}2 \\
1 \\
\end{bmatrix}
$$
The question asks me to find the geometric int... | What you have is the parametric equation for a line. Do you see why?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1010108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If the normalizer of a subgroup in a group is equal to the subgroup then is the subgroup abelian? If $H$ is a proper subgroup of $G$ such that $H=N_G(H)$ ( the normalizer of $H$ in $G$ ) , then is it true that $H$ is abelian ?
| Since no one posted a simple counterexample yet here it goes:
Consider $G=S_4$ and $H=S_3$ considered as a subgroup of $G$ (i.e. as the stabilizer of one of the $4$ points $G$ naturally acts on). Then there are $4$ conjugates of $H$ (the stabilizers of the $4$ points are all conjugate) but also $4=[G:H]$ hence $N_G(H)=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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positive fractions, denominator 4, difference equals quotient (4,2) are the only positive integers whose difference is equal to their quotient. Find the sum of two positive fractions, in their lowest terms, whose denominators are 4 that also share this same property.
| Let's work with $n$ instead of 4. We achieve $a=\frac{b^2}{b-n}=b+n+\frac{n^2}{b-n}$, hence $b-n$ must divide $n^2$. If $a/b$ must be in lowest terms, this is only possible if $b=n+1$: would $b$ be a proper divisor of $n^2$ it would contain a proper divisor $k$ of $n$. Now if $k$ is a proper divisor of $n$ it is a pr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Subtle Twist on the Monty Hall Problem---Does It Make a Difference? In the Monty Hall problem, when the host picks a door and reveals an goat, does it make any difference if he did not know which door the real car was behind, and he just happened to pick a door with a goat?
| Ok, ill turn my comment into a full answer:
Let $C$ be the event where you choose the car and let $G$ be the event that Monty Hall shows the goat. Then the probability that the other door has a car given that monty shows you a goat is:
$P(\neg C|G)=\frac{P(\neg C)(P(G|\neg C)}{P(\neg C)(P(G|\neg C) + P(C)P(G|C)}=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1010434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the angle between a vector and a line defined?
Is the angle between a vector and a line defined?
The angle between two lines $a,b$ is defined as the smallest of the angles created.
The angle between two vectors $\vec{a},\vec{b}$ is the smallest angle one of them has to be rotated by so that the directions of $\vec... | You could just take a vector pointing along the direction of the line and use the angle between the original vector and this vector.
Of course, you would need to choose the orientation your new vector should point: this will affect the angle you get. I would expect you would choose the one which minimizes the angle.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Properties possessed by $H , G/H$ but not G i) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are abelian but $G$ is not ?
ii) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are cyclic but $G$ is not ?
| Sure, here's an answer to both. Consider the quaternions, $Q_{8}$, and the normal subgroup $H = \langle i \rangle$. Then $Q_{8}/H \cong \mathbb{Z}_{2}$, but $Q_{8}$ is neither cyclic nor abelian.
Here's another more general example. Consider $D_{2n} = \langle r, s \mid r^{n}=s^{2} =1, rs=sr^{-1}\rangle$. The subgroup $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1010637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding multiple functions with same $f_{even}$ but different $f_{odd}$? A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$.
If we know only $f_{even}$, how can we find different values for $f_{odd}$ that work (we can't just... | There is a one to one correspondence between $A\times B=\{\text{even}\}\times \{\text{odd}\}$ and $C=\{\text{functions}\}$, given by
$$
F(a,b) = a+b
\\
G(f) = \left( x\to \frac{f(x) + f(-x)}2,x\to \frac{f(x) - f(-x)}2
\right)
$$
In linear algebra terms, $C = A\bigoplus B$.
In other words, $f_{even}$ is not informative... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Some questions about synthetic differential geometry I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a theory? Why does Kock use [[ ]] rather than { } for sets? Does it serve ... | Any elementary topos comes with an internal language which allows you to formally import constructive logic and some set-theoretical notions into it. This enables you to manipulate objects (and arrows) inside it as though they were concrete sets. This is no formal coincidence: the notion of elementary topos was distill... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solve easy sums with Binomial Coefficient How do we get to the following results:
$$\sum_{i=0}^n 2^{-i} {n \choose i} = \left(\frac{3}{2}\right)^n$$
and
$$\sum_{i=0}^n 2^{-3i} {n \choose i} = \left(\frac{9}{8}\right)^n.$$
I guess I could prove it by induction. But is there an easy way to derive it?
Thank you very much... | For the first one consider the binomial expansion of $(1+\frac{1}{2})^n$ and see how close that is the left hand side while adding the values will give the right hand side.
For the second, consider putting in $\frac{1}{8}$ and note what fraction is 2 to the negative 3.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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showing an operator is normal If I have that H is an inner product space with inner product $( . , . )$ over the complex numbers, and $T∈L(H,H)$. Let $R=T+T^*$, $S=T-T^*$ .
If I suppose T is normal, how do I show that :
1) $T^*$ is normal and
2) $R∘S=T∘T-T^*∘T^*$
I'm having trouble even getting started on this proble... | Definitions:
*
*T normal <=> TT* = T*T
*T* normal <=> T*T** = T**T*
Furthermore we have the following properties of the adjoint:
*
*T** = T
*(R + S)* = R* + S*
can you take it from there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1011025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If the pot can contain twice as many in volume, is it twice as heavy? Suppose that I have two pots that look like similar cylinders (e.g. Mason jars). I know that one of them can contain twice as many in volume than the other. If both are empty, is the bigger one twice as heavy as the other?
Intuitively I would say tha... | No, it is not twice as heavy. The weight of the pot is determined by its mass, which is proportionate to its surface, rather than its volume.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1011104",
"timestamp": "2023-03-29T00:00:00",
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Can anyone find out an counter example to this unitary statement? If $A$ and $B$ are unitary, then $A+B$ is not unitary.
I think this statement is true .
I tried to find out counter example but I failed.
| The $0 \times 0$ matrix is unitary, thus it is a counterexample.
Exluding degenerate matrices, $1 \times 1$ matrices are the simplest; they're usually a good place to look for counterexamples, if you aren't trying to exploit noncommutativity.
A $1 \times 1$ unitary matrix is simply a complex number of absolute value $1... | {
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Proving the harmonic series, multiplied by a factor of 1/n, decreases monotically to zero. I would like to show that $\left(1 + \frac 12 + \dots + \frac 1n\right) \cdot \left(\frac 1n\right)$ decreases monotonically to zero.
I have seen one method: to first show that the difference $\left(1 + \frac 12 + \dots + \frac 1... | This is the arithmetic mean of the numbers $1, 1/2,\ldots,1/n$. If you add a number that is lesser than the others, say $1/(n+1)$, the mean decreases.
The limit is $0$ because
$$\frac{H_n}n<\frac{\log n}n\to0$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Why does dividing by zero give us no answer whatsoever? I've heard about this and I know that division can be used in one way like this:
For example, if I want to do $30$ divided by $3$, how many times can I subtract $3$ from $30$ to get to $0$? Well, I can do it this way: $30-3=27-3=24-3=21-3=18-3=15-3=12-3=9-3=6-3=3... | It is an interesting attempt, but you rather cannot induce, that any number $A$ divided by 0 is infinity. We know, that $-0=0$ and from your considerations $A$ divided by 0 must be equal to $-\infty$ in the same time. Moreover, your attempt gives $0/0=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Categorial description of the subposet of $\prod_{i \in I}P_i$ of all $x \in \prod_{i \in I}P_i$ with $\{i \in I \mid x_i = \bot_{P_i}\}$ cofinite By a grounded poset, I mean a poset $P$ with a least element $\bot_P$.
Definition. Whenever $P$ is a family of grounded posets, write $\bigotimes_{i \in I}P_i$ for the subpo... | Your $\bigoplus_{i \in I} P_i$ is the colimit of all finite products of some of the $P_i$; in symbols, $\bigoplus_{i \in I} P_i =\mathop{\mathrm{colim}}_{F \subset I, |F|<\infty} \prod_{i \in F} P_i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1011736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $y=-e^x\cos2x$, show that $\frac{d^2y}{dx^2}=5e^x\sin(2x+\tan^{-1}(\frac{3}{4}))$
If $y=-e^x\cos2x$, show that $\frac{d^2y}{dx^2}=5e^x\sin(2x+\alpha)$ where $\alpha=\tan^{-1}(\frac{3}{4})$.
I've managed to figure out that
$$
\frac{d^2y}{dx^2}=e^x(4\sin2x+3\cos2x)
$$
But I'm not sure how I can massage it into the f... | $4\sin2x+3\cos2x=5(\frac{4}{5}\sin2x+\frac{3}{5}\cos2x)$
Note we want $\frac{4}{5}\sin2x+\frac{3}{5}\cos2x=\sin(2x+\alpha)$ for some $\alpha$.
By compound angle formula, we know $$\sin(2x+\alpha)=\sin2x\cos(\alpha)+\cos2x\sin(\alpha)$$
So in order to fulfill the requirement, we only need to set $$\cos(\alpha)=\frac{4}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1011859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Group homomorphism, the uniqueness of k for g' = gk Group homomorphism is $ \Phi: G \rightarrow H $
Show, that for all $ h \in H $ and all $ g,g' \in \Phi^{-1}(\{h\}) $ there exists a unique $k \in \ker(\Phi) $, so that $g'=gk$.
$$
\forall h \in H, \forall g,g' \in \Phi^{-1}(\{h\})\exists! k \in \ker(\Phi): g'=gk
$$
I... | Notice that $k=g^{-1}g'$ is in $\ker \Phi$ and $g'=gk$ and by the unicity of its form $k$ is unique
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1011960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Give all values of h for which the matrix A fails to be invertible Can someone please help me out with this please. I know the answer is h = 8 and I know the determinant is 21h - 168 and I even know the steps to find those answers. For some reason this is giving me fits and I must be making silly mathematical mistake... | \begin{align*}
\operatorname{det}\begin{bmatrix} 7 & -5 & 3 \\ 14 & -7 & 1 \\ 7 & -8 & h \end{bmatrix}
&= 3\operatorname{det}\begin{bmatrix} 14 & -7 \\ 7 & -8 \end{bmatrix} - 1\operatorname{det}\begin{bmatrix} 7 & -5 \\ 7 & -8 \end{bmatrix} + h\operatorname{det}\begin{bmatrix} 7 & -5 \\ 14 & -7\end{bmatrix} \\
&= 3(14... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Reference textbook about proof techniques I am looking for some good recommended reference textbooks about proof techniques.
Someone told me "G. Polya - How to solve it" is kind of standard, but quite old.
I am looking for a book that handles both classical (manual work) proofs and modern proof techniques using proof a... | I personally recommend Problem Solving Strategies
and Putnam and Beyond.
These are math-olympiad oriented. But I think they are excellent sources and they include loads of examples that are certainly not your average textbook problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
An example for conditional expectation A factory has produced n robots, each of which is faulty with probability $\phi$. To each robot a test is applied which detects the faulty (if present) with probability $\delta$. Let X be the number of faulty robots, and Y the number detected as faulty.
Assuming the usual indenpen... | Denote with $F$ the event that a robot is faulty, with $P(F)=\phi$ and with $T$ the event that it was tested faulty with $$P(T|F)=\delta$$ and $$P(T|F')=0$$ Thus according to the law of Total Probability the probability that a random robot is tested faulty is equal to $$P(T)=P(T|F)P(F)+P(T|F')P(F')=\delta\cdot\phi+0=\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Clarification: Proof of the quotient rule for sequences My Problem
I am currently looking for a proof for the quotient rule for sequences:
$a_n$ and $b_n$ are two sequences with the limes a,b. So:
When
$ a_n \rightarrow a$
and
$ b_n \rightarrow b$
Then:
$\frac {a_n}{b_n} \rightarrow \frac{a}{b}$
Awesome stuff, but how... | If, for eny $\epsilon > 0 \in \mathbb{R}$, there exists an $n \in \mathbb{Z}$ such that $\forall i > n, |a_i - c| < \epsilon $, then $c$ is defined as the limit of the sequence of $a_i$.
Assuming the limit of sequence $a_i$ is $a > 0$, and the limit of $b_i$ is $b > 0$, we want to prove the limit of $\dfrac{a_i}{b_i} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does a vector space need to be closed? What part of the definition of a vector space (see here) requires it to be closed under addition and multiplication by a scalar in the field? I would understand if we defined a vector space as a group of vectors rather then a set but we don't, also non of the axioms require this t... | Because, addition and multiplication (by a scalar) operations are functions from $V \times V$ to $V$ and $K \times V$ to $V$, respectively.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that a positive polynomial function can be written as the squares of two polynomial functions
Let $f(x)$ be a polynomial function with real coefficients such that $f(x)\geq 0 \;\forall x\in\Bbb R$. Prove that there exist polynomials $A(x),B(x)$ with real coeficients such that $f(x)=A^2(x)+B^2(x)\;\forall x\in\Bb... | Survey article by Bruce Reznick called Some Concrete Aspects of Hilbert's 17th Problem, includes your case in the paragraph on Before 1900:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 3,
"answer_id": 1
} |
Limit cycle of dynamical system $x'=xy^2-x-y$, $y'=y^3+x-y$ Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$,
$$
f(x,y)=(xy^2-x-y,y^3+x-y).
$$
Using polar coordinates, one can get
$$
r'=r(r^2\sin^2\theta-1),\quad \theta'=1.
$$
With Mathematica one can get
As one can see from the figure, there is a limit cycle f... | Using the change of variables $(u,v)=(x+y,y)$, the $(x,y)$-differential system is equivalent to the $(u,v)$-differential system $$u'=uv^2-2v,\qquad v'=v^3+u-2v$$ In particular, $$(u^2+2v^2)'=2(uu'+2vv')=2v^2(u^2+2v^2-4)$$
This shows that the ellipsis $(E)$ of equation $u^2+2v^2=4$ is invariant by the dynamics. In the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
Is there a real number $r$ such that $\sum\limits_{k=0}^{\infty}\frac{p_k}{r^k}=e$? Let $p_n$ denote the sequence of prime numbers, with $p_0=2$.
I'm looking for a real number $r$ such that $\sum\limits_{k=0}^{\infty}\frac{p_k}{r^k}=e$.
It's easy to show that $r>5$, with $\frac{2}{5^0}+\frac{3}{5^1}+\frac{5}{5^2}=2.8>e... | Write $f(x) = \sum_k p_k x^k$. Assume first that the series for $f(1/4)$ converges. Then the function $f(x)$ is defined and continuous on $[0,1/4)$. You've shown $f(1/5) > e$, and clearly $f(0) = 2$. By the intermediate value theorem, there must be some $c \in (0,1/5)$ such that $f(c) = e$. You can take $r = 1/c$.
It r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1012940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Approximation of $f\in L_p$ with simple function $f_n\in L_p$ Let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit$$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$$where $\{f_n\}$ is a sequence of simple, i.e. taking countably (infinitely ... | In many sources, simple functions are those measurable functions that have a finite set of values. But here a countably infinite set of values is allowed. This allows one easily approximate any measurable function $f$ uniformly by simple functions: for example, let
$$
f_n(x) = \frac{\lfloor n f(x)\rfloor }{n}
$$
and o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve to find $y(x)$ of the $\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$ Solve the equation to find the $y$ as a function to respect $x$ without $n$ $$\frac{1}{\sum_{n=0}^{\infty }y^n}-\sum_{n=0}^{\infty }x^n=0$$
| Assuming $|x| < 1$ and $|y| < 1$, the two geometric series simplify to:
$$\frac{1}{\frac{1}{1 - y}} = \frac{1}{1 - x}.$$
Consequently:
$$(1 - x)(1 - y) = 1.$$
Solving for $y$ in terms of $x$, you get:
$$1 - y = \frac{1}{1 - x}$$
which implies that
$$y(x) = 1 - \frac{1}{1 - x} = \frac{x}{x - 1}.$$
Note that $|y(x)| < 1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find coordinates of reflected point? How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis?
Example Question:
If P is a reflection (image) of point (3, -3) in the line $2y = x+1$, find the coordinates of Point P.
I know the answer is $(-1,5)$ by drawing ... | Similar answer to @Vrisk, but a bit faster
Consider the line $L=Ax+By+C=0$ and find image of point $(u,v)$ assuming $Au + Bv + C \neq 0 $
If we extend the space which the equation exists into $R^3$ , we will find that the equation $Ax+By+C=0$ denotes a plane with a unit normal vector as : $$ \hat{n} = \frac{<A,B>}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
"answer_id": 4
} |
How to find $P(S_1 \cap S_2^C | K_i)$ given $P(S_1 \cap S_2 | K_i)$ From medical investigations it is known that the symptoms $S_1$ and $S_2$ can appear with three different diseases $K_1, K_2, K_3$. The conditional probabilities $a_{i,j}=P(S_j|K_i), i \in \{1,2,3\}, j \in \{1,2\}$ are given by the following matrix.
$$... | The formula you mention remains true under conditioning, so can be
used:
$P\left(S_{1}\mid K_{i}\right)=\frac{P\left(S_{1}\cap K_{i}\right)}{P\left(K_{i}\right)}=\frac{P\left(S_{1}\cap S_{2}\cap K_{i}\right)}{P\left(K_{i}\right)}+\frac{P\left(S_{1}\cap S_{2}^{c}\cap K_{i}\right)}{P\left(K_{i}\right)}=P\left(S_{1}\cap S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1013347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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