Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Convergence radius and two-times-differentiability of power series. I wanted to compute the radius of convergence for the following the power series
$$\sum_{n=1}^{\infty} a_nz^n$$
with $(i) \, a_n = n!, \, (ii) \, a_n = \sqrt[\leftroot{-3}\uproot{3}n]{n}$
Then I need to determine which power series defines a function t... | Every convergent power series is infinitely differentiable in the same interval of convergence.
$$
f'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}
$$
So radius of convergence of $f'(x)$ is
$$
{1\over\limsup(n|a_n|)^{1\over n}}={1\over\limsup |a_n|^{1\over n}}
$$
as $$\lim_{n\to\infty}n^{1\over n}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ given by $$\tilde{X}=\over... | This really isn't even true when $M$ is a point (this is really the only case I am familiar with). The symplectic blow-up is the natural generalization of the blowup in the complex setting (i.e. the birational isomorphism $\Bbb CP^2 \# \overline {\Bbb CP} ^2 \to \Bbb CP^2$). This is relatively clearly not a fiber bundl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine the number of graphs on the vertex set $\{1, 2, 3 , 4, 5\}$, every vertex is incident to at least one edge. I have the problem of determining how many graphs from the set $\{1, 2, 3, 4, 5\}$ there are, given the property that every vertex is incident to at least one edge. The at least one part of the question... | For $n>0$, the total number of graphs on $n$ vertices is $2^{n(n-1)/2}$. Let $S_k$ be the set of graphs in which vertex $k$ has no incident edge. Then by inclusion/exclusion, the number of graphs you want is
$$|\overline S_1\cap\cdots\cap \overline S_n|
=2^{n(n-1)/2}-|S_1\cup\cdots\cup S_n|
=(-1)^n+\sum_{k=1}^n(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631336",
"timestamp": "2023-03-29T00:00:00",
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Proving positive definiteness of matrix $a_{ij}=\frac{2x_ix_j}{x_i + x_j}$ I'm trying to prove that the matrix with entries $\left\{\frac{2x_ix_j}{x_i + x_j}\right\}_{ij}$ is positive definite for all n, where n is the number of rows/columns.
I was able to prove it for the 2x2 case by showing the determinant is always... | (Update: Some fixes have been added because I solved the problem with $a_{ij}=\frac{2x_ix_j}{x_i+x_j}$ instead of $a_{ij}=\frac{2x_ix_j}{x_i^2+x_j^2}$. Thanks to Paata Ivanisvili for his comment.)
The trick is writing the expression $\frac{1}{x^2+y^2}$ as an integral.
For every nonzero vector $(u_1,\ldots,u_n)$ of real... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Why a code has code words with length $3$ add a parity check matrix become length $4$
Let $\mathcal{C}$ be the code whose codewords are all the words of length 3. Let $D$ be the code formed by adding a parity check matrix digit to each codeword in the code $C$. Find $D$.
The answer key says $D=\{0000, 1001, 0101, 00... | That phrase is surely meant to describe the process (IMO better known) as extending a code by adding an overall parity check bit (or some abbreviated version of that phrase). You begin with a word $x$ of length $n$, and then augment it to a word $x'$ of length $n+1$, where $x'=x|0$ or $x'|1$ and the choice is made in s... | {
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"timestamp": "2023-03-29T00:00:00",
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Upper bound on integral: $\int_1^\infty \frac{dx}{\sqrt{x^3-1}} < 4$ I'm going through Nahin's book Inside Interesting Integrals, and I'm stuck at an early problem, Challenge Problem 1.2: to show that
$$\int_1^\infty \frac{dx}{\sqrt{x^3-1}}$$
exists because there is a finite upper-bound on its value. In particular, sho... | $\int_1^\infty \frac{dx}{\sqrt{x^3-1}}=\int_1^2 + \int_2^\infty=I_1+I_2.$
$I_2 = \int_2^\infty \frac{dx}{x^{3/2}\sqrt{1-1/x^3}}\color{red}{\leq}\lim_{A \to \infty} \int_2^A \frac{dx}{x^{3/2}\sqrt{1-1/A^3}}=\lim_{A \to \infty}\sqrt{\frac{A^3}{A^3-1}}\int_2^A x^{-3/2}dx$
$I_1=\int_1^2\frac{dx}{(x-1)^{1/2}\sqrt{x^2+x+1}}... | {
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What is the average sum of distances of a random point inside a triangle to its three sides? Given a non- Equilateral Triangle with following side sizes: $45,60,75$.
Find the sum of distances from a random located point inside a triangle to its three sides.
Note 1: Viviani's theorem related only to equilateral triangle... | Let $ (x,y)$ be a point in a semi-circle with diameter inclined at $ \sin ^{-1} \frac35 $ to x-axis having sides proportional to Pythagorean triplet (8,6,10) as given.
The three perpendicular distance to sides of a scalene triangle are $ (x,y, 4 x/5 + 3 y/5) , $
totaling to $ \dfrac {9 x + 8 y} {5} $ which is vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation. I have a question.
Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$
I don't know how to proceed. Any help would be appreciated.
| For $\frac1{\alpha^3}+\frac1{\beta^3}$, use that the roots of $a+bx+cx^2$ are $\frac1\alpha$ and $\frac1\beta$ to reduce to the previous problem.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$ How can the following been proved?
$$
\exp(z)\neq0, z\in\mathbb{C}
$$
I tried it a few times, but I failed. Probably it is extremly simple.
If a draw the unit circle and then a complex number $\exp(a+ib)=\exp(a)\exp(ib)$ then it is obvious that th... | If you know that $\exp(z+w)=\exp(z) \exp(w)$, then $\exp(z)\ne 0$ follows from
$$1=\exp(0)=\exp(z-z)=\exp(z)\exp(-z)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1631886",
"timestamp": "2023-03-29T00:00:00",
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Why a truncated table for logic implication $(p\wedge q) \implies p$ verification? The book Discrete Mathematics by Kenneth A. Ross says: "Let's verify the logic implication $(p\wedge q) \implies p$. For that, we need to consider only just the case when $p\wedge q$ is true; i.e., both, $p$ and $q$, are true. This gives... | Recall that an implication $R\implies S$ is "automatically true" in cases where the hypothesis $R$ is false. So the only interesting cases to check are those for which the hypothesis is true.
In this case the hypothesis is $p \wedge q$, so you can proceed just with analyzing what would be implied by that being true.
| {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate the triple integral $\iiint (x^2+y^2+z^2)\,dx\,dy\,dz$ I have to evaluate this integral. It is enough for me to know the correct limits to integration.
$$ \iiint_W (x^2 + y^2 + z^2) \,\mathrm dx\,\mathrm dy\,\mathrm dz$$
Conditions:
$$x\ge 0,\quad y \ge 0 ,\quad z \ge 0,\quad 0 \le x + y + z \le a,\quad (a>0... | Your integral should be of the form
$$\int_{z_0}^{z_1}\int_{y_0(z)}^{y_1(z)}\int_{x_0(y,z)}^{x_1(y,z)}(x^2+y^2+z^2)\,\mathrm dx\,\mathrm dy\,\mathrm dz $$
By the conditions, we find $0\le z\le a$, and for given $z$, $0\le y\le a-z$, and for given $z$ and $y$ $0\le x\le a-y-z$. So in the end it should be
$$\int_{0}^{a}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$. How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backsla... | HINT: Use the fact that
Between any two rational numbers, we have infinite irrational numbers
and similarly,
Between any two irrational numbers, we have infinite rational numbers.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Solve $z^4+2z^3+3z^2+2z+1 =0$ Solve $z^4+2z^3+3z^2+2z+1 =0$ with $z$: a complex variable.
Attempt at solving the problem:
We divide the polynom by $z^2$ and we get:
$z^2+2z+3+\dfrac{2}{z}+ \dfrac{1}{z^2}=0 $ $ $ We set $w=z+ \dfrac{1}{z}$
We now have $w^2+2w+5=0$
$\bigtriangleup = -16$
Let's find $\omega$ s... |
"I don't know how to find z"
Sure you do!
....
If you are correct in what you have done so far and you have
$z + \frac 1z = w$
And $w_1 = -1-2i$ and $w_2 = -1 + 2i$ then
you need to solve $z +\frac 1z = (-1-2i)$ or $z^2 +(1+2i)z + 1=0$
ANd $z + \frac 1z = (-1+2i)$ or $z^2 + (1-2i)z + 1 = 0$.
Both of which can be solv... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Erdős-Mordell theorem geometry proof
Using the notation of the Erdős-Mordell theorem, prove that $PA \cdot PB \cdot PC \geq \dfrac{R}{2r}(p_a+p_b)(p_b+p_c)(p_c+p_a)$.
The notation of the Erdős Mordell theorem means that $p_a$ for example is the distance from the point $p$ to the side $a$ and $R$ is the circumradius. ... | In order to prove this inequality, it is important that you first prove the identity: $r = 4R\sin\left(\dfrac{A}{2}\right)\sin\left(\dfrac{B}{2}\right)\sin\left(\dfrac{C}{2}\right)$. Thus using Jensen's inequality: $p_c = PA\sin A_1, p_b = PA\sin A_2, A = A_1+A_2\Rightarrow p_c+p_b = PA(\sin A_1+\sin A_2)\leq 2PA\sin\l... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inverse of sum of matrices (SVD, ridge regression) Looking at these slides, I've found the following:
$X=UDV^T$, where $U$ and $V$ are orthogonal matrices, $V$ is a square matrix, and $D$ contains the singular values of $X$.
The author then writes
$$X(X^TX + \lambda I)^{−1}X^Ty$$
$$=UD(D^2 + \lambda I)^{−1}DU^Ty.$$
Why... | Here is the point :
In: $\ UDV^T(VD^2V^T+\lambda I)^{-1}VDU^Ty$
you should write $I=VV^T$ giving the factorization:
$UDV^T(V(D^2+\lambda I)V^T)^{-1}VDU^Ty$
$=UDV^T(V^T)^{-1}(D^2+\lambda I)^{-1}V^{-1}VDU^Ty$
yielding the looked-for formula.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intuition behid $P(A\mid B)$. What is the intuition behind the formula $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$
I have seen this formula around, but every site/book I look at does not really have a clear & cut explanation behind this formula.
| It's equivalent to $$\frac{n(A\cap{B})}{n(B)}$$ in other words the proportion of the members of set $B$ which are also members of set $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1632697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Conditional Expectation - using Wald's Equation
Let $N\sim\!\mathcal{P}(\lambda)$ and $(X_i)_{i\!\geq{1}}$ iid, $X_i\sim\!Be(p)$. If $N$ and $(X_i)_{i\!\geq{1}}$ are independent for all $i$, calculate $P(\mathbb{E}(X_1+\ldots+X_N|N)=0)$.
So using Wald's equation and the fact that $(X_i)_{i\!\geq{1}}$ are iid, I know... | As you found, $\mathsf E\left(\sum\limits_{j=1}^N X_j\;\middle\vert\; N\right)= Np$
You know that $N\sim\mathcal P(\lambda)$ so you can find $\mathsf P(Np{=}0)$ from
$$\mathsf P(N{=}k) \;=\; \dfrac{\lambda^k\, {\sf e}^{-\lambda}}{k!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1632799",
"timestamp": "2023-03-29T00:00:00",
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Find the sum $\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$ Find the sum $$\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$$
When I asked my teacher how can I solve this question he responded it is very hard, you can't solve it. I hope you can help me in solv... | I am re-editing a-rodin's answer, correcting a few typos [of an earlier version, now edited].
\begin{align}
\sum\limits_{k=1}^{100} \frac {k\cdot k!}{100^k} \frac{100!}{k!(100-k)!} &= \frac{100!}{100^{100}} \sum\limits_{k=1}^{100} \frac{k\cdot100^{100-k}}{(100-k)!}\\
&= \frac{100!}{100^{100}} \sum\limits_{k=0}^{99}\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1632928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 2
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Is the punctured plane homotopy equivalent to the circle? I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$.
However, $X$ and $Y$ are not homotopic, i.e. we can't find continuous maps $f:X\to Y, g : Y \to X$ such t... | Just to answer this question: they are homotopy equivalent. If $g : Y \to X$ is the natural embedding, and $f$ is as above, then
$g \circ f$ is homotopy equivalent to $id_X$ :
Let $H_1 : X \times [0,1] \to X$ be defined as
$$(x, t) \mapsto tx/\|x\| + (1-t)x.$$
(Even if $X$ is not convex, $H_1(x,t) \in X$ for any valu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1633029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Dense on the unit circle I am reading:
"It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)"
How is this possible? Can anyone give me an intuitive explanation of this?
(Not a solution)
Thank you.
| Intuitive explanation. Let $\alpha$ be an irrational angle - not a rational multiple of $2\pi$. Then if you start at $(1,0)$ and step around the unit circle in steps of size $\alpha$ the set of points you reach will come as close as you like to any other point. That follows from the fact that you can never get back to ... | {
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"question_score": "2",
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Matrix induction proof Given the following
$\lambda_{1}=\frac{1-\sqrt{5}}{2}$ and $\lambda_{2}=\frac{1+\sqrt{5}}{2}$
How do I prove this using induction:
$\begin{align*}
A^k=\frac{1}{\sqrt{5}}\left(\begin{array}{cc}
\lambda_2^{k-1}-\lambda_1^{k-1} & \lambda_2^{k}-\lambda_1^{k}\\
\lambda_2^{k}-\lambda_1^{k} & \lambda_2^... | 1) A straightforward proof which is more natural than recursion, in my opinion (for a recursion proof see 2).)
Use diagonalization identity $A=P\Lambda P^{-1}$ from which $A^k=P\Lambda^kP^{-1} \ \ (1)$
where $\Lambda$ is the diagonal matrix diag$(\lambda_1,\lambda_2)$.
Here is an extension of my first explanation:
Inde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1633222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Detailed balance implies time reversibility, how about the converse? Given a Markov chain (finite state space) $X_1,X_2,...$ with transition matrix $P$ and initial distribution $\pi$, if they satisfy $\pi(x)P(x,y)=\pi(y)P(y,x)$, we say they satisfy detailed balance.
If the joint distribution of $(X_1,X_2,...,X_n)$ is i... | $\Bbb P[X_0=x_0,X_1=x_1]=\pi(x_0)P(x_0,x_1)$ and $\Bbb P[X_0=x_1,X_1=x_0]=\pi(x_1)P(x_1,x_0)$, so the $n=1$ case of reversibility already implies detailed balance.
| {
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Confused about proof by contradition In proof by contradiction, I can understand how it works when the hypothesis leads to a clearly false proposition. e.g., if we want to prove $P$, we assume $\neg P$ and show that $\neg P \implies ... \implies Q$, but we know that $\neg Q$ and since we just proved that $\neg Q \impli... | Let $Q$ be the statement $P\land\lnot P$. We know that $Q$ is false.
Then if you have $\lnot P\implies P$ then you have $\lnot P\implies (P\land \lnot P)$, which is $P\implies Q$. But we know $Q$ is false, so $P$ is false.
Indeed, every proof by contradiction can be written as:
$$P\implies(A\land \lnot A)$$
for some p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1633419",
"timestamp": "2023-03-29T00:00:00",
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How to minimize $ab + bc + ca$ given $a^2 + b^2 + c^2 = 1$? The question is to prove that $ab + bc + ca$ lies in between $-1$ and $1$, given that $a^2 + b^2 + c^2 = 1$.
I could prove the maxima by the following approach. I changed the coordinates to spherical coordinates:
$a = \cos A \\
b = \sin A \cos B \\
c = \sin A ... | From what you've done, $ab+bc+ca = \dfrac{(a+b+c)^2 - (a^2+b^2+c^2)}{2} \geq \dfrac{0 - 1}{2} = \dfrac{-1}{2}$, and this is the minimum value you sought. The minimum occurs when $a+b+c = 0, a^2+b^2+c^2 = 1$. To solve for $a,b,c$ you only need to find one solution of the system of $2$ equations above, then you are done.... | {
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Can you find the maximum or minimum of an equation without calculus? Without using calculus is it possible to find provably and exactly the maximum value
or the minimum value of a quadratic equation
$$ y:=ax^2+bx+c $$
(and also without completing the square)?
I'd love to know the answer.
| One approach for finding the maximum value of $y$ for $y=ax^2+bx+c$ would be to see how large $y$ can be before the equation has no solution for $x$.
First rearrange the equation into a standard form:
$ax^2+bx+c-y=0$
Now solving for $x$ in terms of $y$ using the quadratic formula gives:
$x= \frac{-b\pm \sqrt{b^2-4a(c-y... | {
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Calculating the length of the paper on a toilet paper roll Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with small error) the total paper length of a toilet roll?
Writin... | Lets do the spiral version.
Using your notation, a spiral joining circles of radiuses $r$ and $R$ with $N$ twists has the form $S(t)=(r+\frac{tb}{2\pi N})e^{i t}$, where $t\in[0,2\pi N]$
The length $L$ of the spiral is
$$\begin{align}
L & = \int_{0}^{2\pi N}|S'(t)|dt \\
& = \int_{0}^{2\pi N}\Big|\frac{b}{2\pi N}e^{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1633704",
"timestamp": "2023-03-29T00:00:00",
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What is the order when doing $x^{y^z}$ and why? Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why?
Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$?
I always get confused with this and I don't understand the underlying rule. Any help would be appreciated!
| The exponent is evaluated first if it is an expression. Examples are $3^{x+1}=3^{\left(x+1\right)}$ and $e^{5x^3+8x^2+5x+10}$ (the exponent is a cubic polynomial) and $10^{0+0+0+10^{15}+0+0+0}=10^{10^{15}}$. The left-associativity simply fails when the exponent contains multiple terms.
| {
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"timestamp": "2023-03-29T00:00:00",
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Limit on a five term polynomial it has been two years since I have taken or used calculus, and I am having some trouble with factoring a polynomial in order to take a limit on it. I have searched for previous similar questions here, but I have been unable to find anything helpful. Here is my problem:
I have:
\begin{equ... | Ok, so I figured out my error. I was allowing the term \begin{equation}13x^2\end{equation} from the numerator to mess me up. Factoring the numerator needs to be approached by first turning the prime number, 13, into non-prime numbers. Thus,\begin{equation}x^4+3x^3-13x^2-27x+36\end{equation}becomes \begin{equation}x^4+3... | {
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Cantor's Intersection Theorem with closed sets Cantor's Intersection Theorem states that "if $\{C_k\}$ is a sequence of non-empty, closed and bounded sets satisfying $C_1 \supset C_2 \supset C_3 \dots$, then $\bigcap_{n \ge 1} C_n$ is nonempty.
If the term "compact sets" is replaced by "closed sets", the statement is n... | Consider the sequence $C_n = [n, \infty)$.
| {
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Defining natural transformations based on generalized elements? Let $F : \mathbf{C} \to \mathbf{D} : G$ be two functors between categories $\mathbf{C}$ and $\mathbf{D}$. A natural transformation $\eta$ from $F$ to $G$ is a collection of morphisms $\eta : FC \to GC$ in $\mathbf{D}$ for each $C \in \mathbf{C}$.
Particula... | What Awodey is trying to express in intuitive notation is that the twist map is $\langle \pi_1, \pi_0\rangle:A\times B\to B\times A$, so that $\pi_0\circ t_{(A,B)}=\pi_1$ and $\pi_1\circ t_{(A,B)}=\pi_0$. It's easy to see that for any pair of generalized elements $a:Z\to A$ and $b:Z\to B$ this will give you the propert... | {
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Find $\lim_\limits{R\to \infty}{1\over 2\pi i}\int_{{1\over 2}-iR}^{{1\over 2}+iR}{x^s\over s}ds$ $\lim_\limits{R\to \infty}{1\over 2\pi i}\int_{{1\over 2}-iR}^{{1\over 2}+iR}{x^s\over s}ds$ where $x>0$. Split it to cases: $x>1,x=1,0<x<1$. I tried using contour integration but I am still very confused as to how I shoul... | Rewrite as
$$\lim_{R \to \infty} \frac1{i 2 \pi}\int_{\frac12-i R}^{\frac12+i R} ds \frac{e^{s \log{x}}}{s} $$
When $x \gt 1$, $\log{x} \gt 0$ and we may close to the left of $\operatorname{Re}{s}=\frac12$ with a circle of radius $R$ centered at the origin. The integral about the circle vanishes when $R \to \infty$. ... | {
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Guide to solving Harary's exercises Most of Harary's harder exercises are research problems (although solved), that need almost always a single key idea as a breakthrough. Often it so happens that even after thinking for a long time no solution comes to mind.
I am a beginner in Graph Theory. How should I solve the pro... | Choose vertices $a$ and $u$ at maximum distance. There is one neighbor of $a$ at distance $D-1$ from $u$. What is the distance from $u$ of the other neighbors? Working from here, that $a$ and $v$ have the same valency.
Now consider a cycle of length $2D+1$, and prove that all vertices on it have the same valency.
To be... | {
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"source": "stackexchange",
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prove homomorphic image has order 4 Let $G$ be a group of order $20$ show that $G$ has homomorphic image of order $4$.
From Cauchy theorem we have elements $a,b \in G$ of orders $2,5$ respectively then
from first isomorphic theorem homomorphic image is isomorphic with $G/kerf$ for some homomorphism $f:G \to im(f)$
so... | $G/\langle b\rangle$ is such a homomorphic image. The subgroup generated by $b$ is normal by the Sylow's theorems. Indeed the number $n$ of subgroups of order $5$ is congruent to $1\mod 5$, and a divisor of $20/5=4$. Hence $n=1$, which proves the subgroup is normal.
| {
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Why it is more accurate to evaluate $x^2-y^2$ as $(x+y)(x-y)$ in floating point system? The expression $x^2-y^2$ exhibits catastrophic cancellation if $|x|\approx|y|$.
Why it is more accurate to evaluate as $(x+y)(x-y)$ in floating point system (like IEEE 754)? I see this is intuitively true. Any one can help demonstr... | I guess the answer is when $x$ and $y$ are large, $x^2$ and $y^2$ are larger. If they are of the same order of magnitude then when you consider evaluating $x^2 - y^2$ you are taking away two large numbers and the round off error will be larger than if you write it is $(x + y) (x - y)$. For $x \approx y$, the second app... | {
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In $\lim(x,y)\to(0,0)$ why can I change to $(x^2,x)$? When we have a multivariable function and we want to see if the function is continuous at a point, normally the origin, we sometimes "change" $(x,y)\to(0,0)$ to expressions like $(x^2,x)\to(0,0)$ to make it work.
For example, for the function:
$f(x)=\begin{cases}\... | Confronted with such a problem you have to make a decision, founded on your experience with similar problems: Shall I try to prove that the limit exists, or shall I try to prove that the limit does not exist?
If you conjecture that the limit $\lim_{{\bf z}\to{\bf 0}}f({\bf z})$ does not exist you can try to exhibit two... | {
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Notation conversion help with respect to combinatorical proof First off, I wouldn't bring this to SO, but my teacher refuses to teach me notation.
Anyhow...
I'm doing a proof. The proof says:
"Show that $8^n - 3^n$ is a multiple of 5 for all non-negative integers $n$.
How do I say this in notation?
I've got: $\f... | First, let me say that your teacher might refuse teching you this notation because it is considered bad style in written mathematics. The symbols from formal logic (like $\forall$) should be used nearly exclusively when talking about formulas in formal logic and maybe (carefully!) as a shorthand on the blackboard. Full... | {
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Evaluating $\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$ I am trying to find$$\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$$
But I can't figure out any good way to solve this.
Is there a special theorem or method to solve such limits?
| Hint
$$\ln(n+2)-\ln(n) = \ln\bigg (\frac{n+2}{n}\bigg)$$
| {
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Convergence of $\sum_{n=1}^\infty \sqrt[n]{2}-1$ I'm trying to determine whether $$\sum_{n=1}^\infty \left ( \sqrt[n]{2}-1\right )$$ converges or diverges. Ratio, root, nth term, etc tests are either inconclusive or too difficult to simplify. I feel like there must be something I can bound this series by but I can't ... | Hint:
$$\sqrt[n]{2}-1 = e^{\frac{\log{2}}{n}} - 1 = \frac{\log{2}}{n} + O(n^{-2})$$
as $n \to \infty$.
ADDENDUM
To address your specific problem, consider instead
$$\left (1+\frac{\log{2}}{n} \right )^n $$
which you should be able to show is less than $2$. I that case, then you can show that $\sqrt[n]{2}-1 \gt (\log{2... | {
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Show that for $p \neq 2$ not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square. Show that for $p\neq2$, not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square of an element in $\mathbb{Z}/p\mathbb{Z}$. (Hint: $1^2=(p-1)^2=1$. Deduce the desired conclusion by counting).
So far I have that $1=p^2-2p-1\Rightarrow p... | Because $\mathbb{Z}/p\mathbb{Z}$ is finite the map $f \colon \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$, $x \mapsto x^2$ is surjective (i.e. every element is a square) if and only if it is injective. But for $p \neq 2$ we have $-1 \neq 1$ with $f(-1) = 1 = f(1)$.
| {
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The nature of roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0,$ If the expression $ax^2+2bx+c$, where $a$ is a non-zero real number, has the same sign as that of $a$ for every real value of $x$, then roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0$ are:
(A) real and equal
(B) real and unequal
(C) non-real h... | Since $ax^2+2bx+c$ has always the same sign as $a$ for any real $x$, it has no real roots, so $4b^2 - 4ac < 0$.
Now try writing
\begin{align}
(b-c)^2 + 4a(2b+c+a) &= (b-c)^2 + 4(2ab+ac+a^2) \\
&= (b-c)^2 + 4(2ab+b^2+a^2) + 4(ac - b^2) .
\end{align}
| {
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When is the rank of Jacobian constant? Suppose I've got a function $f : \mathbb{R}^{n} \to \mathbb{R}^{m}$ which I know is bijective.
Considering $\mathcal{J}$, the Jacobian of $\ f$, I want to understand what can be said about the rank of $\mathcal{J}(\mathbf{x})$.
Let's say I evaluate $\mathcal{J}(\mathbf{0})$, and ... | If $f$ is a bijection, the Jacobian need not even exist. Let $m=n$ and $f$ be a generic permutation of $\Bbb{R}^m$. A random permutation is unlikely to be continuous anywhere, much less have derivatives.
(Bi-)Continuity (being homeomorphic) isn't sufficient to guarantee a derivative exists. Consider continuous nowhe... | {
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Distance function is continuous in topology induced by the metric The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ defined by $f(x) = d(a,x)$ is continuous.
My first attem... | Not sure why you use $a$ both for elements in $\mathbb R$ and in $X$. Note by triangle inequality,
$$d(x, a)\le d(x, y)+ d(y, a)\Rightarrow d(a, x) - d(a, y)\le d(x, y)$$
Interchanging the role of $x, y$ and use $f(x) = d(a, x)$ we have
$$\tag{1}|f(x) - f(y)|\le d(x, y).$$
This inequality is sufficient for us to cond... | {
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If $G$ is of order $n$ and $k$ is prime to $n$ then $g(x)=x^k$ is one-to-one
Let $G$ be a group of order $n$ and let $k$ be prime to $n$, show that $g(x) = x^k$ is one-to-one.
I started trying to prove this, and said:
If $g(x) = g(y)$ for some $x,y \in G$ then $x^k = y^k$. Also, since $n,k$ are prime we have that th... | Since $G$ is finite, you only have to show injectivity. So let $x \in G$ with $x^k=1$. Then the order of $x$ is a divisor of $k$. But we also know, that the order of $x$ is a divisor of $n$. Hence the order of $x$ is a common divisor of $k$ and $n$, hence equal to $1$.
| {
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Find a thousand natural numbers such that their sum equals their product The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach :
I worked on this question for lesser cases :
\begin{align}
&2 \times 2 = 2 + 2\\
&2 \times 3 \times 1 = 2 + 3 + 1\\
&3 \times 3 \time... | There's a sign error in your final equation; you want
$$ a+b+998=ab $$
which simplifies to
$$ (a-1)(b-1) = 999 $$
from which it should be easy to extract several integer solutions.
| {
"language": "en",
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Amount of interest Find the amount of interest earned between time t and n where $t<n$; if $I_r=r$ for some positive integer r. Answer is $\frac{1}{2}(n^2+n-t^2-t)$
$I_{[t,n]}=A(n)-A(t)$
$I_{[0,r]}=A(r)-A(0)=r$
$A(r)=A(O)+r$
For $t<r<n$
$I_{[r,n]}=A(n)-A(r)$ ;(1)
$I_{[t,r]}=A(r)-A(t)$ ;(2)
Cannot grasp any further conc... | $$
A(n)-A(t)=\sum_{r=t+1}^n I_{[0,r] }=\sum_{r=t+1}^n r = \sum_{r=1}^n r-\sum_{r=1}^t r=\frac{n(n + 1)}{2}-\frac{t(t + 1)}{2}=\frac{1}{2}(n^2 + n- t^2 - t)
$$
| {
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Sum of all elements in congruence class modulo n With $+$ defined as $[a]+[b]=[a+b]$, show that $[0]+[1]+\cdots+[n-1]$ is equal to either $[0]$ or $[n/2]$ in $\Bbb Z_n$.
How do I go about proving this? I have managed to get $[(n^2-n)/2]$ using the definition but how do I proceed from here to the result?
Help would be ... | If $n=2k+1$ is odd, $\left[\dfrac{n(n-1)}2\right]=[nk]=[0]$.
If $n=2k$ is even, $\left[\dfrac{n(n-1)}2\right]=[k(n-1)]=[-k]$.
| {
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BMO2 2016 Number Theory Problem Suppose that $p$ is a prime number and that there are different positive
integers $u$ and $v$ such that $p^2$ is the mean of $u^2$ and $v^2$. Prove that
$2p−u−v$ is a square or twice a square.
Can anyone find a proof. I can't really see any way to approach the problem.
| Note that $2p^2=u^2+v^2$, or $(p-u)(p+u)=(v-p)(v+p)$. WLOG, suppose $u<p<v$.
From the above equation, we have:
$$2p-u-v=(p-u)+(p-v)=\frac{(v-p)(v-u)}{p+u}$$
Now, we do following analysis:
If $q$ is odd prime, and $q^a|(v-p)$ then $q^a\not|(v+p)$ since $p$ is prime, and $p\not|v$.
So, $q^a|(p-u)(p+u)$, and only one of... | {
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Infimal convolution $g^\star = f_1^\star + f_2^\star$ Let $f_1$ and $f_2$ be convex functions on $R^n$. Their infimal convolution $g = f_1 \diamond f_2$ is defined as
$$
g(x) = \inf \{f_1(x_1) + f_2(x_2) \mid x_1 + x_2 = x\}.
$$
Prove that $g^\star = f_1^\star + f_2^\star$.
| I finally found a solution to the problem
$$
g*(y) = \underset{x}{\sup}\; \{x^Ty - g(x)\}
$$
As we know
$$
g(y) = \underset{x_1 + x_2 = x}{\inf}\; \{f_1(x_1) + f_2(x_2)\}
$$
Now we have
$$
g*(y) = \underset{x}{\sup}\; \{x^Ty - \underset{x_1 + x_2 = x}{\inf}\; \{f_1(x_1) + f_2(x_2)\} \}
$$
$$
= \underset{x}{\sup}\;... | {
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u-substitution, indefinite integrals I've looked on the web for an answer to this question, and could not find an example.
Could you push me towards a proper u substitution for the following integral? Please don't solve the problem just state what you would use as a substitution and why.
$$\int(\sin^{10}x \cdot \cos x)... | thanks to u/justpassingthrough
$$\int(sin^{10}x*cosx)dx$$
let $$u=sinx$$
$$du=cosx$$
$$\int(sin^{10}x*cosx)dx=\int(u^{10})du$$
$$=\frac{1}{11}u^{11}+C=\frac{1}{11}(sinx)^{11}+C$$
cool
| {
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Polynomial divides set of points Given a set of points in the plane with distinct $x$-coordinates, each point colored black or white. A polynomial $P(x)$ "divides" the set of points if no black point lies above $P(x)$ and no white point lies below $P(x)$, or vice versa. Points of any color can lie on $P(x)$.
What is th... | For all $n$, there is a way to choose the points & colors that forces $ P $ to have degree at least $ n-2 $ .
Assume $ n \geq 3 $ (the smaller cases are easy).
Choose the points $ \{(i,0) | 1 \leq i \leq n-2\} \cup \{(n-1,1),(n,n^2)\}$. Color them alternating white and black in order of their $ x $-coordinates.
For an... | {
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How to find proper functions to bound for integral squeeze thereom I am trying to prove that the function on $[0,1]$ defined by $$f(x)=\begin{cases} 1 & \text{if $x=\frac{1}{n}$ , $n \in \mathbb{N}$}\\ 0 &\text{else} \end{cases}$$
Is Riemann integrable on this interval. I would like to use squeeze theorem but I am hav... | Let's choose a sequence of functions $f_N$, which is $1$ in little $\delta$ intervals around those $\frac{1}{n}$ with $n \leq N$, and is $1$ for $x \in [0, \frac{1}{N}]$. Then you have exactly $N$ little $\delta$ balls and a little remainder.
Note that the integral of $f_N$ is always at least as large as the integral o... | {
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Sum of Linearly Dependent Vectors Is it possible that vectors $v_1, v_2, v_3$ are linearly dependent, but the
vectors $w_1=v_1+v_2$, $w_2=v_1+v_3$, $w_3=v_2+v_3$ are linearly independent? I
believe the answer is no, this is not possible, but I am struggling to formulate
the proof.
| No, they must be dependent. Note that you have
$$\begin{align}
v_1 &= \frac{1}{2}(w_1+w_2-w_3) \\
v_2 &= \frac{1}{2}(w_1-w_2-w_3) \\
v_3 &= \frac{1}{2}(-w_1+w_2-w_3)
\end{align}$$
Now, by assumption the $v_i$'s linearly dependent, so that there exist $(a,b,c)\neq(0,0,0)$ such that $0=av_1 + bv_2+c v_3$. This gives, if... | {
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An example of a reducible random walk on groups? Random walk on group is defined in the following way as a Markov chain.
A theorem says the uniform distribution is stationary for all random walk on groups.
If the random walk is irreducible, for example, on a cyclic group $\Bbb Z_n$, then the stationary distributi... | It is not quite true that any random walk on $\mathbb{Z}_n$ is irreducible. This will depend on the increment distribution $\mu$ as well.
For example, on $\mathbb{Z}_4$ if you take $\mu(g)=\delta_{g+2}$ then there are two communicating classes $\{0,2\}$ and $\{1,3\}.$ There are lots of different stationary distributio... | {
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Is there a formula for the area under $\tanh(x)$? I understand trigonometry but I've never used hyperbolic functions before.
Is there a formula for the area under $\tanh(x)$?
I've looked on Wikipedia and Wolfram but they don't say if there's a formula or not.
I tried to work it out myself and I got this far:
$\tanh(x... | Notice, we know $\sinh(x)=\frac{e^x-e^{-x}}{2}$ & $\sinh(x)=\frac{e^x+e^{-x}}{2}$, hence the area under $\tanh(x)$ is
$$\int \tanh(x)\ dx=\int \frac{\sinh(x)}{\cosh (x)}\ dx$$
$$=\int \frac{e^x-e^{-x}}{e^x+e^{-x}}\ dx$$
$$=\int \frac{d(e^x+e^{-x})}{e^x+e^{-x}}$$
let $e^x+e^{-x}=t\implies d(e^x+e^{-x})=dt $,
$$=\int \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1637179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Contour Integration: non-convergent integral The question is
$$I=\int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} dx$$
My attempt:
$$I=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{e^{2ix}-2+e^{-2ix}}{x^2} dx$$
$$I=-\frac{1}{4} \Big[ \int_{-\infty}^{\infty} \frac{e^{2ix}}{x^2} dx - \int_{-\infty}^{\infty} \frac{2}{x^2} dx +\... | We can use Parseval's theorem to solve the integral.
The rectangle function is,
$$\Pi(x) = \begin{cases} 1 \quad |x|<1/2 \\ 0 \quad \text{otherwise} \end{cases}$$
The Fourier transform of the rectangle function is the $\mathrm{sinc}$ function.
$$ \hat{\Pi}(k) = \mathrm{sinc}(k) = \frac{1}{\sqrt{2\pi}}\frac{\sin(k)}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the value of $x$ which is correct I have one exercise which is $$(x+2013)(x+2014)(x+2015)(x+2016)+1=0$$
I tag $A=x+2013$ or other for many ways but still can not find the first $x$ value. please help.
| If you let $y=x+2014$, then the equation becomes
$$(y-1)y(y+1)(y+2)+1=0 \Leftrightarrow (y^2+y-1)^2=0$$
So
$$y = \frac{-1\pm\sqrt{5}}{2} \Rightarrow x = \ldots$$
\begin{align}
(y-1)y(y+1)(y+2)+1 & = y^4+2 y^3-y^2-2 y+1 \\
{} & = \left( y^4+y^3-y^2 \right) + y^3-2y+1 \\
{} & = y^2 \left( y^2+y-1 \right) + \left(y^3+y^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1637361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it true that for any set $X,Y \subseteq \Bbb A^n(k)$, $I(X) = I(Y )$ iff $X = Y$ . Is it true that for any set $X,Y \subseteq \Bbb A^n(k)$, $I(X) = I(Y )$ iff $X = Y$ . Because I know $I(X) = I(Y )$ implies $\overline X=\overline Y$ But I am thinking that this is not true because if I take $Y=\overline X$ then $I(X)... | The correspondence you mentioned, $X = Y$ if and only if $I(X) = I(Y)$, is only valid when $X, Y$ are assumed to be closed subsets of $\mathbb{A}^n(k)$. In the proposition you mentioned, all the sets which they were dealing with were closed, so there is no trouble there.
For an example when $I(X) = I(Y)$ holds, but $X... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is the commutator subgroup of a profinite group closed? Let $G$ be a profinite group, $[G,G]=\{ghg^{-1}h^{-1}|g,h\in G\}$ is a subgroup of $G$. Is $[G,G]$ closed?
In the case we are interested, $G$ is the absolute galois group of a local field.
| In general is not true that for a profinite group $G$ the derived subgroup $G'$ is closed; But if $G$ is a pro-$p$-group, with $p$ a prime, and if $G$ is finitely generated then it becomes true. For example this is true for the group of $p$-adic integer $\mathbb{Z}_{p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1637512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Image of a family of circles under $w = 1/z$ Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of $z$, $\overline{z}$, and then plug in $z = 1/w$.
However, I am having difficu... | Write $z=x+iy$, so $x^2+y^2=z\bar{z}$, and $x=\frac{z+\bar{z}}{2}$. Thus the circles can be described by
$$
z\bar{z}=a\frac{z+\bar{z}}{2}
$$
Upon doing $z=1/w$, you get
$$
\frac{1}{w\bar{w}}=\frac{a}{2}\frac{\bar{w}+w}{w\bar{w}}
$$
that becomes
$$
a\frac{\bar{w}+w}{2}=1
$$
Writing $w=X+iY$, you get
$$
aX=1
$$
More gen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1637621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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If $f(x)$ is continuous at $ x=0$ Given that $f(x)$ is continuous at $ x=0$, and the limit : $$\lim \limits_{x \to 0} \frac{f(x)}{x^2} = L$$
then:
$$\implies f(0) = 0 $$
and
$$ \implies f(x) \text{ is differentiable at }x=0 $$
My question is: why $f(0) = 0 $ ?
| Given $\varepsilon >0$ there exist $\delta_{\varepsilon}>0$ such that
$$-\delta_{\varepsilon}<x<\delta_{\varepsilon}\qquad \implies \qquad\left|\frac{f(x)}{x^2}-L\right|<\varepsilon$$
So $$|f(x)-Lx^2|<\varepsilon x^2$$
Now, take $\delta=\min(\delta_{\varepsilon},1)$, then
$$-\delta<x<\delta\qquad \implies \qquad\left|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate $\lim \limits_{h \to 0}{\frac{a^h-1}{h}}$? As the title says, I would like to prove for $f(x) = a^x$ there is always some constant c such that $f'=cf$. Is calculating the limit the right approach to solve this problem? Also, how to show there is only one solution when $c=1$? (the $e^x$)
| Assuming $a>0$, you need to use the definition of real exponentiation:
$$a^h=e^{h\ln a}=\sum_{n=0}^\infty\frac{(h\ln a)^n}{n!}$$
Then
$$\lim_{h\to0}\frac{a^h-1}h=\lim_{h\to 0}\sum_{n=1}^\infty h^{n-1}\frac{(\ln a)^n}{n!}=\ln a$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find a parametric equation to solve a line integral Find
$ \int_C \frac{dz}{z} $ from $1 - 5i$ to $5 + 6i$
I want to use
$$\int_C f(Z) dz = \int_a^b f[z(t)]z'(t)dt$$
My guess is that I need to find $z(t)$ such that
$$z(a) = 1 - 5i$$
$$z(b) = 5 + 6i$$
But how?
| You can actually use any path from $a$ to $b$. You should get the same answer as long as the path does not include the origin. This is the Principle of Path Independence. It follows from the Cauchy-Goursat Theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$ Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$.
I know how to prove that $U\cap W$ is a subspace o... | "Closed under addition" means that $u+w \in U$ would be true if $u,w \in U.$ Written more concisely, $$U + U \subseteq U.$$ It's not enough when $u \in U$ and $w$ isn't. This is why your proof doesn't work.
You can make a proof around the computation $$(U+W) + (U+W) = (U+U) + (W+W) \subseteq U + W.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1638221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Integral of $-4\sin(2t - (pi/2)) $ weird behavior on wolfram alpha I'm confused by what Wolfram Alpha is doing with my function:
$$-4\sin{(2t - (\pi/2))}$$
on why the it gets replaced by
$$4\cos{(2t)}$$.
Is it equal? Link: See behavior here
| Yes. Using the angle-addition formula,
$$ \sin{(2t-\pi/2)} = \sin{2t}\cos{(\pi/2)}-\cos{2t}\sin{(\pi/2)} = -\cos{2t}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1638300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integration by Parts? - Variable Manipulation
$$\int x^3f''(x^2)\,\mathrm{d}x$$ Solve using Integration by Parts.
\begin{align}
u&=x^3\qquad\mathrm{d}v=f''(x^2) \\
\mathrm{d}u&=3x^2\qquad v=f'(x^2) \\
&=x^3f'(x)-\int f'(x^2)3x^2 \\
u&=3x^2\quad\mathrm{d}v=f'(x^2) \\
\mathrm{d}u&=6x \qquad v=f(x^2) \\
&=x^3f'(x^2)-[3... | I suggest you start with a small change of variable (just to make life easier) since $$I=\int x^3f''(x^2)~dx=\frac 12\int x^2 f''(x) 2x dx$$ S0, let $x^2=y$, $2x\,dx=dy$ which make $$I=\frac 12\int y\,f''(y) \, dy$$ Now, one integration by parts $$u=y\implies u'=dy$$ $$dv=f''(y)dy\implies v=f'(y)$$ So $$2I=u\, v-\int v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1638407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Kelly criterion for Each-Way betting 3 outcome answered question
Hi all,
I've been having trouble finding the Kelly Criterion bet size for an each-way bet.
The above link shows the solution to a problem with 3 distinct and mutually exclusive outcomes. In an each-way bet there are three distinct outcomes (Lose all bets,... | The question to which you link to, answers your question too, as you can treat these as 3 different outcomes.
For example, if you back a horse E/W with x-x @5/1 odds with 1/5 E/W for places 1-2-3, then, in David Speyer's answer, you need to substitute for no place $b_1=-2$, for place 2-3 $b_2=1-1=0$, and for win $b_3=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1638638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Isomorphic to Subgroup of even permutations True or False
Every finite group of odd order is isomorphic to a subgroup of $An$, the group of all even permutations.
The question was in entrance exam. I think there is counter example to this statement but i am not reaching that example. Can some one help?
| One can embed $S_n$ can be embedded into$S_n\times S_n$ diagonally, i.e., $\sigma\mapsto (\sigma,\sigma)$ we see that $S_n$ embeds into $A_{2n}$ and so every finite group, order is odd or even, can be embedded into suitable alternating group.
(Compare it with similar statement any matrix group can be embedded into $SL(... | {
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"timestamp": "2023-03-29T00:00:00",
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Is $C_0(\mathbb{R})$ a Banach space? Let $C(\mathbb{R})$ be a Banach space of continuous real-valued functions defined on $\mathbb{R}$, with supremum norm, and let $C_0(\mathbb R)$ be the subspace of functions vanishing at infinity. Is $C_0(\mathbb{R})$ a Banach space?
I try to see it using:
$f\in C_0(\mathbb{R})$ iff... | $C_0(\mathbb R)$ is a Banach space because it may be identified with a closed subspace of some $C(K)$, the real vector space of continuous functions on the compact Hausdorff space $K$, equipped with the $\|\cdot\|_\infty$ norm. Since uniform convergence on compact sets is so well-behaved, $C(K)$ is a Banach space (belo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fixed points for 1-D ODE I'm doing some independent work, and have managed to come across the following interesting 1-D autonomous ODE:
$\dot{x} = x(1-x) \log^2\left[\frac{x}{1-x}\right]$.
For the fixed points, i.e., where $\dot{x} = 0$, I know that the only valid one should be for $x = 1/2$ because of the Log function... | Writing
\begin{equation}
f(x) = x(1-x) \log^2 \frac{x}{1-x},
\end{equation}
the limits $\lim_{x\to 0} f(x)$ and $\lim{x \to 1} f(x)$ both exist and yield 0. As you noticed, the singularity in the log is 'overpowered' by the fact that $x(1-x)$, which is polynomial, has roots at $x = 0$ and $x = 1$. So, $x = 0$ and $x =... | {
"language": "en",
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Calculate area of figure
What is the area of the shaded region of the given $8 \times 5$ rectangle?
Attempt: We know that the area of a kite is $\dfrac{pq}{2}$ where $p$ and $q$ are the diagonals. Thus, since the two lines in the figure intersect at the center of the square, the length of the long diagonal of one of... | We draw a diagonal from the upper-left to the bottom-right corner. This divides the two quadrilaterals into four triangles, two with base $1$ and height $4$ and two with base $1$ and height $\frac{5}{2}.$ Using basic area calculations, the total area should be $\boxed{6.5}.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Computation of an iterated integral I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not integrable, I can't use the Theorem of Change of Variable. So, I'm trying to use residue form... | Let $D$ be any Jordan domain in $\mathbb{R}^2$, containing origin in its interior,
whose boundary $\partial D$ has the form $r = f(\theta)$ in polar coordinates where
$f \in C[0,2\pi]$.
Consider following integral as a functional of $D$:
$$\mathcal{I}_D \stackrel{def}{=} \int_D \phi(x,y) dx dy
\quad\text{ where }\quad\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Marginalising out $B$ in $P(A \mid B,C)$ Let's say that I have $P(A \mid B,C)$ - is it accurate to say that $P(A \mid C)$ can be found like this:
$P(A \mid C) = \sum_B P(A \mid B,C)$
I know the values of all $P(A \mid B,C)$ as well as $P(B)$ and $P(C)$
| You probably know (marginalization, or total probability)
$$P(A) = \sum_B P(A,B)$$
Because this is true for any $A,B$, it's also true for the variables conditioned on $C$ (which amounts to restricting the universe).
$$P(A \mid C) = \sum_B P(A,B\mid C)$$
Further $P(A,B\mid C)=P(A \mid B,C)P(B\mid C)$.
(Why? Because $P... | {
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"timestamp": "2023-03-29T00:00:00",
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$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$ I'd appreciate if you could please check to see if my proof is valid.
Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$.
Solution:
$q^2 > p^2/5 \iff q > p/\sqrt{5} \iff 0<p/q<\sqrt{5} $.
$\implies E = \mathbb{Q}... | There is a mistake in part (ii): as pointed out in the comments, $0$ is not in $E$.
What you wrote in part (i) is true, but more complicated than it needs to be. Here are a few suggestions to simplify your proof:
*
*Delete the first sentence. The variable $M$ is not used anywhere in the rest of the proof.
*Delete t... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit in integral: $\lim\limits_{\epsilon \rightarrow 0}\frac1{\epsilon}\int_{t}^{t+\epsilon}f(s)ds=f(t)$? Let $f$ be a smooth function :
can someone tell me why we have :
$\lim\limits_{\epsilon \rightarrow 0}\frac{1}{\epsilon}\int_{t}^{t+\epsilon}f(s)ds=f(t)$
thank you very much !
| Let $F(x) = \int_0^x f$. Then $F$ is smooth since $f$ is, and $F^\prime = f$ by the Fundamental Theorem of Calculus; but we also have $$F^\prime(t) = \lim_{\varepsilon\to0}\frac{F(t+\varepsilon) - F(t)}{\varepsilon} = \lim_{\varepsilon\to0}\frac{\int_t^{t+\varepsilon} f}{\varepsilon}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate the derivative of logarithm of a matrix? Given a square matrix $M$, we know the exponential of $M$ is
$$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$
and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$
The derivative of $\exp(M)$ should be itself. It is easy to prove if $\frac{dM}{M}=I$.... | The derivative of $\log(x)$ is $1/x$. The derivative of the power series
$$ \sum_{n=1}^\infty (1-x)^n/n$$ is $$\sum_{n=1}^\infty (1-x)^{n-1}$$ which converges to $x^{-1}$ if $|x| < 1$.
The matrix power series $$\sum_{n=1}^\infty (I-M)^{n-1}$$ converges to $M^{-1}$ if the spectral radius of $I-M$ is less than $1$.
Howe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the sequence of combinations contains an odd number of odd numbers
Let $n$ be an odd integer more than one. Prove that the sequence $$\binom{n}{1}, \binom{n}{2}, \ldots,\binom{n}{\frac{n-1}{2}}$$
contains an odd number of odd numbers.
I tried writing out the combination form as $$\frac{(2k+1)!}{(m!)((2k+... | Suppose $n=2k+1$
Note that $\binom{2k+1}{1}=\binom{2k+1}{2k}$, $\binom{2k+1}{2}=\binom{2k+1}{2k-1}$,...,$\binom{2k+1}{k}=\binom{2k+1}{k+1}$.
Thus
$$\binom{2k+1}{1}+\binom{2k+1}{2}+\cdots+\binom{2k+1}{k}+\binom{2k+1}{k+1}+\cdots+\binom{2k+1}{2k-1}+\binom{2k+1}{2k}=2^{2k+1}-2$$
From the above considerations,
$$2\binom{... | {
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"timestamp": "2023-03-29T00:00:00",
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About Factorization I have some issues understanding factorization.
If I have the expression $x^{2}-x-7$ then (I was told like this) I can put this expression equal to zero and then find the solutions with the quadratic formula, so it gives me $x_{0,1}= 1 \pm 2\sqrt{2}$ then $$x^{2}-x-7 = (x-1-2\sqrt{2})(x-1+2\sqrt{2})... | $3(x - 1)(x + {2 \over 3}) = (x -1)(3x + 2) = (3x - 3)(x + {2 \over 3}) =...$ etc. are all valid factoring. The leading coefficient is just a constant.
And if $(x - 1)(x + {2 \over 3}) = 0$ then $3(x - 1)(x + {2 \over 3}) = 0 = (x - 1)(x + {2 \over 3}) $.
If you are concerned about going from roots to factoring think... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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In the context of ordered statistics, each of Y(1),Y(2),...,Y(n) a single observation or distributions that are I.I.D? In statistics one aspect of the I.I.D. concept that bothers is when I think about it in the context of ordered statistics. As most of you already know, $Y_1,Y_2,Y_3,...,Y_n$ are I.I.D. when the paramet... | If you have $Y_1,Y_2, \dotsc, Y_n$ and each are independent and follow some distribution $G$, then you could consider each $Y_i$ as a realization, or sample, taken from $G$.
If you then order, then each $Y_{(i)}$ follows a new distribution.
For example, say we have $Y_1, \dotsc, Y_n$, where each one is independent an... | {
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Countability of generated ring $R(E)$ I am studying Paul R. Halmos Measure theory. In the section 5 of chapter 1, theorem 5 states that :
If $E$ is a countable class of sets, then $R(E)$ is countable.
The proof uses class of all finite unions of differences of sets of class E.
Can anyone explain this is a simple manne... | Let $X = \bigcup E$ be the set of which all members of $E$ are subsets. For good measure, assume that $X\in E$ (if it isn't, use $E'=E\cup {X} in what follows).
Note that $R(E)$ is closed under intersection: if $A,B\in R(E)$, then $(X\setminus A)\cup (X\setminus B) = X\setminus(A\cap B) \in R(E)$, so $A\cap B = X\setm... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to solve $\int \frac{1}{1-y^2}$ with respect to $y$? I was solving an A Level paper when I came across this question. I tried substitution, but I'm not getting the answer with that. Would appreciate it if someone would help me.
| Using a simple substition of $y = \tanh{(u)}$ gives
$\displaystyle\int \dfrac{1}{1-y^{2}} \mathrm{d}y = u + C = \tanh^{-1}y + C = \ln{\dfrac{1 + y}{1 - y}} + C$
Notably can also be solved using partial fractions.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $|A| < |B|$ if $A-B$ is positive definite? I want to prove this. Say if $A-B$ is a positive definite matrix then can we find a relation between $\det(A)$ and $\det(B)$?
e.g. is $|A| < |B|$.
| Consider $A=2$, $B=1$. Then $A-B=1$ is positive definite but det$(A)=2 \not<$ det$(B)=1$. If you have mistyped the direction of the $<$ sign for the determinants then with:
$$A=\begin{pmatrix}1& 0 \\ 0& 1\end{pmatrix}\qquad B=\begin{pmatrix}-2& 0 \\ 0& -2\end{pmatrix}$$
$$A-B=\begin{pmatrix}3& 0 \\ 0& 3\end{pmatrix}$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range I was studying the Inverse function theorem when I came across the following problems :
(Let the closed set $V$ i.e the range have non-empty interior)
*
*Does there exist a continuous onto... | The sine/cosine functions map $\mathbb{R}$ to the closed interval $[-1,1]$.
(THIS IS WRONG: Moreover, the embedding $\mathbb{R} \to \mathbb{R}^2, x \mapsto (x, 0)$ satisfies both assertions.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1640409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Inverse of the composition of two functions If I have a composition of two functions:
$$y = f(g(x),h(x))$$
where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I know this is generally possible for the the composition of one function.
Perhaps... | EDIT: Previous answer used the accepted definition of 'composition'. I am trying to interpret what the OP means by composition... I think he means that $f$ is a formula e.g. something like $f(\sin x,e^x)=\sin x\sqrt{e^x}$ means that $f(x,y)=x\sqrt{y}$.
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $\frac{1}{\sin x}-\frac{1}{x}$ uniformly continuous on $(0,1)$? So I am tasked with finding whether $\frac{1}{\sin(x)}-\frac{1}{x}$ is uniformly continuous on the open interval $I=(0,1)$. To look at the "simple" ways to prove it first:
I obviously can't extend the function to $[0,1]$ since the limit at $x=0$ is not ... | As $x>\sin x$ in $(0,1)$ $$f'(x)<{1\over1+\cos x}<1\;\forall\; x\in(0,1)$$
Just replace $\approx$ with $<$. You don't need that $\lim_{x\to0}{\sin x\over x}=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1640624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$? If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$?
The problem is the following:
In the solution of Folland chapter 1 exercise 14,
Suppose $F^*=${$F:... | Let $$\mathcal{F} = \{F\subset E: F \ \ \text{is measurable and} \ 0 < \mu(F) < \infty\}$$
Since $\mu$ is semi-finite, $\mathcal{F}$ is not empty. Let
$$s = \sup\{\mu(F):F\in\mathcal{F}\}$$
It suffices to show that $s = \infty$. Choose a sequence $\{F_n\}_{n\in\mathbb{N}}$ such that $\lim_{n\to \infty}\mu(F_n) = s$. T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Counting permutations with given condition I need to find number of permutations $p$ of set $\lbrace 1,2,3, \ldots, n \rbrace$ such for all $i$ $p_{i+1} \neq p_i + 1$.
I think that inclusion-exclusion principle would be useful. Let $A_k$ be set of all permutation that for every permutation $a$ in this set $a_{k+1} \neq... | We can complete Penitent's induction argument as follows:
Denote the number of valid permutations of $k+1$ elements by $a_k$. There are two ways to generate a valid permutation of $k+1$ elements from a permutation of $k$ elements: Either by inserting $k+1$ into a valid permutation of $k$ elements at any of the $k+1$ ga... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove if $\sum\limits_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\sum\limits_{n=1}^ \infty a_nb_n$ converges. Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.
No, $a_n, b_n$ are not nec... | Outline: Indeed, the key is use Dirichlet's test (a.k.a. Abel's summation at its core) as you intended:
$$\begin{align}
\sum_{n=1}^N a_n b_n &= \sum_{n=1}^N (A_n-A_{n-1}) b_n =
A_Nb_N + \sum_{n=1}^{N-1} A_n b_n -\sum_{n=1}^{N-1} A_n b_{n+1} \\
&= A_Nb_N + \sum_{n=1}^{N-1} \underbrace{A_n}_{\text{bounded}} \underbrace{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Joint distributions where one is uniform
Let $X$ have a uniform distribution on the interval $(0,1)$.
a) Find the c.d.f. and p.d.f. of $Y=\dfrac{X}{1-X}$.
b) Find the c.d.f. and p.d.f. of $W=\ln Y$.
I am extremely confused on part A, and part B also. I get to this very early step and am stuck:
$$Y= \dfrac{X}{1-X} \\... | We have
$$F_Y(y)=\Pr(Y\le y)=\Pr\left(\frac{X}{1-X}\le y\right).$$
This is $\Pr(X\le y(1-X))$, which is $\Pr(X(1+y)\le y)$. Finally, for $y$ positive, which is the only interesting part, we have
$$F_Y(y)=\Pr\left(X\le \frac{y}{1+y}\right)=\frac{y}{1+y}.$$
Elsewhere, we have $F_Y(y)=0$.
For the density function, differ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$? Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that without loss of generality it c... | $$10181^4 + 1 = 2 \cdot 17 \cdot 1657 \cdot 4657 \cdot 5113 \cdot 8009$$
I have many factoring variants. Many of the commands in the following involve string variables, these are used in the versions where the factorization will actually be printed out, but are irrelevant here.
part of file form.h
#include <iostr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 1
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How to determine the reflection point on an ellipse Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point P is reflected by the ellipse and arrivates at Q. The question is how to determine th... | Hint:
We can use a method similar to that of Max Payne, but representing the ellipse by its parametric equations $x = a \cos t$, $y = b \sin t$. By differentiation, the tangent vector is $(- a \sin t, b \cos t)$, hence the normal vector $(b \cos t, a \sin t)$.
Now we express the equality of the angles to the normal, by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Is it true that $ f(n) = O(g(n))$ implies $g(n) = O(f(n))$ So I have this is an assignment for algorithms. I've googled a lot, read the chapter in the book about big Oh notation, and I understand the concept. I do not however understand how to prove it. I know I need to work with a constant and put that out of the equa... | The definition of the big-oh notation is as follows :
$f(x)=O(g(x))$ if $|f(x)| \leq c|g(x)|$ for every big enough $x$ and some constant $c$
This is why $f(x)=x$ and $g(x)=x^2$ is a counter-example :
*
*$x=O(x^2)$ because for example taking $c=1$ we have $x \leq x^2$ for every $x \geq 1$
*$x^2$ can't be $O(x)$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How did the rule of addition come to be and why does it give the correct answer when compared empirically? I'm still a high school student and very interested in maths but none of my school books describe these kind of things, and only say how to do it not the whys. My question is very simple, for example:
19
... | A visual representation:
19
+ 25
= 44
Let's remove the answer and split the numbers into tens and units:
10 + 9 {= 19}
+ 20 + 5 {= 25}
This give us:
30 + 14
We can make a new addition out of these numbers:
14
+ 30
Let's split these numbers into tens and units too:
10 + 4 {= 14}
+ 30 + 0 {= 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 16,
"answer_id": 11
} |
What does third derivative tell about inflection point? I was trying to find the nature (maxima, minima, inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$
But I faced a conceptual problem. It is given in the solution to the problem that $f''(0)=0$ and $f'''(0) \neq 0$ so $0$ is not an inflection ... | It depends on your definition of inflection point. You have given this as
"Where curve changes its concavity".
In this case checking $f'''(x)$ is not necessary since for example,
$$f(x) = x^3 \text{ gives } f'''(x) = 6 \neq 0 $$ however $x = 0$ is an inflection point since there is a change in concavity here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way…
If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is an eigen vector of $B... | The key is to realize that you multiply partitioned matrices together in exactly the same manner by you do "regular" matrix matrix multiplication. The only difference is that multiplication is no longer commutative because your are dealing with submatrices rather than scalars. I shall demonstrate. Let $A$ and $B$ be pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove this integral is analytic Let $\phi$ be a continuous (complex valued) function on the real interval [−1, 1] inside C, and define
$$f(z)=\int_{-1}^1\frac{\phi(t)}{t-z}dt$$
Show that f is analytic on C less the interval [−1, 1].
I thought about CR equation to prove analytic but the function is not of the form $f(z) ... | If you wanted to use CR:
(This answer only works for $\phi$ real, but if $\phi=\phi_1+i\phi_2$ with $\phi_1,\phi_2$ real, you can see that it follows from the real case.)
Write $z=a+bi$ then $$\frac{\phi(t)}{(t-a)-bi} = \frac{\phi(t)(t-a +bi)}{(t-a)^2+b^2}$$
So $$\int_{-1}^{1} \frac{\phi(t)}{t-z}\,dt = \int_{-1}^1 \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,... I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a convergent subsequence of $x_n$ with limit $c$. Then, we let... | The error is in the sentence that starts "It seems to me...." The set of cluster points for the sequence you describe is, in fact, all of $\mathbb{R}$, not just $\mathbb{Q}$, for pretty much precisely the reason you give in the next paragraph: for any real number, consider a sequence of rationals converging to that n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1641992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Product of two primitive roots $\bmod p$ cannot be a primitive root. I recently proved that the product of all primitive roots of an odd prime $p$ is $\pm 1$ as an exercise. As a result, I became interested in how few distinct primitive roots need to be multiplied to guarantee a non-primitive product. Testing some smal... | A primitive root of an odd prime $p$ must be a quadratic non-residue of $p$, and the product of two non-residues is a residue.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Can I further simplify $5^k \cdot 5 + 9 < 6^k \cdot 6$ to prove this is true I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive.
$k ≥ 2$
Can I further simplify this?
| Hint:
It's not hard to check that $6^{k+1}-{5}^{k+1}$ is increasing and $(6\cdot6-5\cdot 5) \geq 9$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
limit of $f(x) = \lim \limits_{x \to 0} (\frac{\sin x}{x})^{\frac 1x}$ Any ideas how to calculate this limit without using taylor?
$$f(x) = \lim \limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac1x}$$
| Take the log of both sides and examining the 2 sides of the limits yields
$$\ln L^+=\lim \limits_{x \to 0^+} \frac{\ln (\sin x)-\ln(x)}{x}$$
$$\ln L^-=\lim \limits_{x \to 0^-} \frac{\ln (\sin (-x))-\ln(-x)}{x}$$
which can be solved by L'Hopital's to both equal $0$, so the limit is 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Arc length of a sequence of semicircles. Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$.
Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one joining $0,1$ below the $x$-axis, call this curve $\gamma_2$
Keep subdividing the ... | The problem with this argument, and many other alike, is that the arc length of a curve is not a continuous function of a curve, which means that if sequence of curves $\gamma_i$ converges to a curve $\gamma$ (notion of convergence of functions like that can be made formal, but let me not do that here), then there is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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