Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof that every number ≥ $8$ can be represented by a sum of fives and threes. Can you check if my proof is right?
Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$.
Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\
x=9 = 3\cdot3 + 5\cdot0 \quad \checkmark\\
x=10 = 3\cdot... | Any number is of the form $5k+1$, $5k+2$, $5k+3$, $5k+4$ and $5k+5$.
If $n=5k+3$ then nothing to prove.
$n=5k+1 \implies n=5(k-1)+2\cdot3$
$n=5k+2 \implies n=5(k-2)+4\cdot3$
$n=5k+4\implies n=5(k-4)+8\cdot3$
$n=5k+5\implies n=5(k-2)+5\cdot 3$
Hence for all $k\ge5$ the result holds.
Hence for all $n\ge 25$ the result h... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
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Prove $7$ divides $13^n- 6^n$ for any positive integer I need to prove $7|13^n-6^n$ for $n$ being any positive integer.
Using induction I have the following:
Base case:
$n=0$: $13^0-6^0 = 1-1 = 0, 7|0$
so, generally you could say:
$7|13^k-6^k , n = k \ge 1$
so, prove the $(k+1)$ situation:
$13^{(k+1)}-6^{(k+1)}$
$... | Write $13=6+7$ to expand $13*13^k-6*6^k$.
| {
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Intersection between two planes and a line? What is the coordinates of the point where the planes: $3x-2y+z-6=0$ and $x+y-2z-8=0$ and the line: $(x, y, z) = (1, 1, -1) + t(5, 1, -1)$ intersects with eachother?
I've tried letting the line where the two planes intersect eachother be equal to the given line, this results ... | From the line equation you know that $x$ (as well as $y$) is a function of $z$:
when $z = -1-t$, then $x = 1+5t$ and hence $x = -4-5z$.
This gives you the third equation you need:
\begin{align}
3x-2y+\phantom{1}z-6&=0\\
\phantom{1}x-\phantom{1}y-2z-8&=0\\
\phantom{1}x+0y+5z+4&=0.
\end{align}
| {
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Stuck when tackling the computation of $\Phi_n(\zeta_8)$ My current way of calculation of $\Phi_n(\zeta_8)$ where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial and $\zeta_8=\cos(\frac{2\pi}{8})+i\sin(\frac{2\pi}{8})$ leave me now stuck at the problem of calculating $$\prod_{d|n}\sin(\frac{d2\pi}{16})^{\mu(\frac{n}{d}... | It's a classic and elementary fact that since $x^n-1=\prod_{d\mid n}\Phi_d(x)$, by applying $\log$s to both sides followed by Mobius inversion and then exponentiating back we must have
$$\Phi_n(x)=\prod_{d\mid n}(x^d-1)^{\mu(n/d)}.$$
This is mentioned in almost any source that covers cyclotomic polynomials. Furthermor... | {
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If people who work at 100% of a program, do the full program in 42 weeks, how many weeks does someone who works at 60% capacity need?
If people who work at 100% of a program, do the full program in 42 weeks, how many weeks does someone who works at 60% capacity need?
I said 75.6 weeks and here's how I got it:
*
*p... | My friend, you yourself have stated that one working at 50% needs 84 weeks. (You got this by multiplying 100/50 and 42)
So similarly, one working at 60% needs 42*(10/6)=70 weeks.
Also, one working at 75% needs 42*(100/75)= 56 weeks; you make the mistake here itself, hence your calculations for 60% are wrong.
| {
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Getting two consecutive $6$'s A standard six-sided die is rolled until two consecutive 6s appear. Find the expected number of rolls. Please see that this is a language problem. What do they mean by expected number of rolls? Should the answer be 36? Or 42?
| Let $E$ be the expected number of rolls. This is the mean number of roll we expect to take to witness the event.
To obtain two consecutive sixes, you must roll until you get a six followed immediately by a six (obviously).
Let the expected number of rolls until you get one six be : $F$. Can you find what $F$ is? (H... | {
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square root / factor problem $(A/B)^{13} - (B/A)^{13}$ Let
$A=\sqrt{13+\sqrt{1}}+\sqrt{13+\sqrt{2}}+\sqrt{13+\sqrt{3}}+\cdots+\sqrt{13+\sqrt{168}}$ and
$B=\sqrt{13-\sqrt{1}}+\sqrt{13-\sqrt{2}}+\sqrt{13-\sqrt{3}}+\cdots+\sqrt{13-\sqrt{168}}$.
Evaluate $(\frac{A}{B})^{13}-(\frac{B}{A})^{13}$.
By Calculator, I have $\frac... | Let
$$A=\sum_{n=1}^{168}\sqrt{13+\sqrt{n}},B=\sum_{n=1}^{168}\sqrt{13-\sqrt{n}}$$
since
$$\sqrt{2}A=\sum_{n=1}^{168}\sqrt{26+2\sqrt{n}}=\sum_{n=1}^{168}\left(\sqrt{13+\sqrt{169-n}}+\sqrt{13-\sqrt{169-n}}\right)=A+B$$ so we have $x=\dfrac{A}{B}=\sqrt{2}$,then we have
$$x=\sqrt{2}+1,\dfrac{1}{x}=\sqrt{2}-1\Longrightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1181905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Behaviour of roots of a polynomial with function coefficients Let $(-1+c_4(h))x^4 +c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)=0$ be an equation with variable coefficients, depending smoothly on $h$. Also let $0\le c_4(h)\le 1-\epsilon$ for some $\epsilon>0$ and $c_0(h)>\epsilon'$ for some $\epsilon'>0$. One can easily see th... | In fact, you can have one or two positive roots. Using the IFT,
$$
0\ne\frac{\partial}{\partial x}(-1+c_4(h))x^4+c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)\Big\vert_{x=z(h)}
$$
$$
=4(-1+c_4(h))x^3+3c_3(h)x^2+2c_2(h)x+c_1(h)x\Big\vert_{x=z(h)}
$$
while
$$(-1+c_4(h))x^4 +c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)\vert_{x=z(h)}=0,$$
i.e.... | {
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Prove $3a^4-4a^3b+b^4\ge0$ . $(a,b\in\mathbb{R})$ $$3a^4-4a^3b+b^4\ge0\ \ (a,b\in\mathbb{R})$$
We must factorize $3a^4-4a^3b+b^4\ge0$ and get an expression with an even power like $(x+y)^2$ and say an expression with an even power can not have a negative value in $\mathbb{R}$.
But I don't know how to factorize it sinc... | If $a=b$, the expression is zero, so take the factor $a-b$:
$$3a^4-4a^3b+b^4=(a-b)(3a^3-a^2b-ab^2-b^3)$$
Again, the cubic is zero if $a=b$:
$$=(a-b)^2(3a^2+2ab+b^2)$$
Now try to show the quadratic is always positive or zero.
| {
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Perpendiculars to two vectors So this is more than likely a simple question that has been asked before, but if have 2 lines described by the formula:
$V = (x,y,z) + L (i,j,k)$
i.e. a line described by a position and a length along a unit vector
How would I find the line that is perpendicular to both lines?
| Hint: Take the cross product.First write each in the form (a,b,c) by choosing t equal 0 and 1.
| {
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What is this motivation for changing variables to standardize a pde? In a question I am looking at it says -
Find a change of variables $u \rightarrow v$ of the form $u = ve^{ax + by}$ that would transform the PDE
$$\frac{\delta^2u}{\delta x^2} + \frac{\delta^2u}{\delta y^2} + \frac{\delta u}{\delta x} + 2\frac{\delta ... | The motivation comes from solving ODE with the method of integration factors. Suppose you're to solve $y'+2y=f(x)$. A natural approach is to let $z=e^{ax}y$ and compute $z'=e^{ax}(y'+ay)$. In order to match the original ODE, we let $a=2$. Then $z'=e^{2x}f(x)$, which we can integrate to solve $z$, and hence $y=e^{-2x}z$... | {
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If both $x+10$ and $x-79$ are perfect squares, what is $x$?
If both $x+10$ and $x-79$ are perfect squares, what is $x$?
That is:
$x+10=$perfect square (or can be square such that the result of it squared is a positive integer)
$x-79=$perfect square (or can be square such that the result of it squared is a positive... | Assuming you are looking for positive integers...
Consecutive squares differ by consecutive odd amounts. The first differences to $\{0,1,4,9,16,\ldots\}$ are $\{1,3,5,7,\ldots\}$.
You have two squares that are $89$ apart. They may or may not be consecutive. But for some string of consecutive odd numbers, the sum must b... | {
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Integration question on $\int \frac{x}{x^2-10x+50} \, dx$ How would I integrate
$$\int \frac{x}{x^2-10x+50} \, dx$$
I am not sure on how to start the problem
| I would complete the square in the denominator first.
$$ \displaystyle \int \dfrac{x}{x^2-10x+25+25}dx = \int \dfrac{x}{(x-5)^2+25}dx $$
Let $u=x-5$,
$$ \displaystyle \int \dfrac{u+5}{u^2+25}du = \int \dfrac{u}{u^2+25}du + \int \dfrac{5}{u^2+25}du$$
the first integral you can solve by doing one more substitution, th... | {
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Hatcher 2.2.26 Show that if $A$ is contractile in $X$ then $H_n(X,A) =H_n(X) \oplus H_{n-1} (A)$
Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$
I know that $\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA) \approx H_n(X,A)$.
And $(X \cup CA)/X = SA$, where $SA$ ... | Hatcher suggests to use the fact that $(X\cup CA)/X\simeq SA$: in order to do that you can consider what you obtained in the point (a) of the exercise.
From (a) you know that $A$ is contractible in $X$ iff $X$ is a retract of $X\cup CA$. Since $X$ is a retract of $X\cup CA$ you have that the following sequence splits:
... | {
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To show every sylow p-subgroup is normal in G Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x)=x$ implies that $x=e$. Prove that for every prime $p$ which divides $o(G)$, the $p$-Sylow subgroup is normal in G.
Its ... | This is not exactly an easy exercise! The proof outline below is from Burnside's book, "The Theory of Groups of Finite Order".
*
*Show that $G = \{x^{-1}\phi(x) : x \in G \}$.
*If $g$ and $\phi(g)$ are conjugate in $G$, then $g=1$. (Proof. Suppose $\phi(g) = h^{-1}gh$. By 1, $h = x^{-1}\phi(x)$ for some $x$, so $\p... | {
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Solving $\lim\limits_{n\to\infty}\frac{n^n}{e^nn!}$ I was solving a convergence of a series and this limit popped up:
$$\lim\limits_{n\to\infty}\frac{n^n}{e^nn!}$$
I needed this limit to be $0$ and it is in fact (according to WolframAlpha), but I just don't see how to get the result.
| Almost as good: write the expression as
$$
L = e^{n \log n - n - \log n!} = e^{n \log n -n -\sum_{k=1}^{n} \log k}
$$
and use the bounds on the sum:
$$
\int_{1}^{n} \log x dx < \sum_{k=1}^{n} \log k < \int_{1}^{n+1} \log x dx
$$
to get the same result without Stirling.
| {
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If a group $G$ is generated by $n$ elements, can every subgroup of $G$ by generated by $\leq n$ elements? I wonder if the statement in the title is true, so if:
$$G=\langle g_1,g_2,\ldots,g_n\rangle$$
means that we can write every subgroup $H$ of $G$ as
$$H=\langle h_1,h_2,\ldots,h_m\rangle$$
for some $h_i\in G$ and $m... | The statement is true for abelian groups: every abelian group generated by $n$ elements is a quotient of $\mathbb{Z}^n$ and every subgroup of $\mathbb{Z}^n$ is free generated by at most $n$ elements. This can be found in any text elementary group theory.
It is false for non abelian groups. For example the permutation g... | {
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Use the definition of differentiation on a piecewise function. I need to find the derivative at $x=0$.
$$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\
0 & \text{if } x \leqslant 0 \end{cases} $$
Using the definition, I know that it's equal to $0$. However, I also need to prove that $f'(x)$ is not continuou... | Indeed $f'(0) = 0$. We have, then: $$f'(x) = \begin{cases} 2x\sin(1/x) - \cos(1/x),& \text{if }x\neq 0 \\ 0, & \text{if } x \leq 0\end{cases}$$
Just check that $\lim_{x \to 0}f'(x)$ does not exist and you are done.
| {
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Orthonormal Basis Proofs Suppose $u, v, w$ are three differentiable functions from $R \to R^3$ such that for every $t ∈ R$ the vectors $u(t), v(t), w(t)$ form an orthonormal basis in $R^3$.
i) Prove that $u′(t) ⊥ u(t)$ and $u′(t)·v(t) = −u(t)·v′(t)$ for all $t$.
I did
$\theta=cos^{-1}((u_1'(t),u_2'(t),u_3'(t))\bullet(u... | $\{u(t),v(t),w(t)\}$ orthonormal basis implies $u(t)\cdot u(t)=1.$ Then
$$0=\frac{d}{dt}1=\frac{d}{dt}(u(t)\cdot u(t))=u'(t)\cdot u(t)+u(t)\cdot u'(t)=2u'(t)\cdot u(t)\implies u'(t)\cdot u(t)=0.$$
$\{u(t),v(t),w(t)\}$ orthonormal basis implies $u(t)\cdot v(t)=0.$ Then
$$0=\frac{d}{dt}0=\frac{d}{dt}(u(t)\cdot v(t))=u'(t... | {
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Certificate of primality based on the order of a primitive root Reading my textbook, it tells me that to prove $n$ is prime, all that is necessary is to find one of its primitive roots and verify that the order of one of these primitive roots is $n-1$.
Now, why exactly does this work? It's not making much sense to me. ... | If there exists a primitive root, this already means that $(\mathbb Z/n)^{\times} \cong \mathbb Z/\varphi(n)$ is cyclic, where $\varphi$ is Euler's totient function. If $\varphi(n) = n-1$, then $n$ is prime.
| {
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Find the coordinate of third point of equilateral triangle. I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates.
I think there will be two possible answers, as the point C could be on the either side of line joining A and B.
Now I put AB = AC =... | If you like vector approach:
Displace (shift/translate) point A by vector(-3,-4) to come to the origin.
Multiply the new radius vector $ab$ with $ e^ {i \pi/3} , e^ {-i \pi/3}$ (once clockwise and once anticlockwise) to obtain new points $ C_1 $ and $ C_2 $. The multiplying factor is $ (1/2 \pm i \sqrt 3/2) $.
Displa... | {
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One-sided nilpotent ideal not in the Jacobson radical? Problem XVII.5a of Lang's Algebra, revised 3rd edition, is:
Suppose $N$ is a two-sided nilpotent ideal of a ring $R$. Show that
$N$ is contained in the Jacobson radical $J: = \{ \cap\, I: I \text{ a
maximal left ideal of } R \}$.
I put my solution below the fo... | That is totally correct.
If you require further validation, then check out Lam's First course in noncommutative rings pg 53, lemma 4.11 which has exactly the generalization you describe, with the same proof.
| {
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What is a "right" automorphism? Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads:
If an automorphism of $F_n$ is an isomorphism from $F_n$ to itself, then what is a right automorphism?
| Automorphisms are special cases of group actions - the automorphism group acts on the underlying set of some first group. In particular this is a left group action. If $A$ is a group and we have a set $X$, recall a left action of $A$ on $X$ is a map $A\times X\to X$ denoted $(a,x)\mapsto ax$ which satisfies the "assoc... | {
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What function satisfies the following equation? $$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$
I think it should be similar to Zeta function, but what is it exactly?
| You are right about the similarity of $f(x)$ to the Riemann Zeta function.
Using the following identity that can be found here
$$\pi^{-\frac{z}{2}}\Gamma\left(\frac{z}{2}\right)\zeta(z)=\pi^{-\frac{1-z}{2}}\Gamma\left(\frac{1-z}{2}\right)\zeta(1-z)$$
and setting $z=\frac{2x}{\pi}$ results in
$$\color{red}{\pi^{-\frac{... | {
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Convergence series problem. How to show that $$\sum_{n=3}^{\infty }\frac{1}{n(\ln n)(\ln \ln n)^p}$$ converges if and only if $p>1$ ?
By integral test,
$$\sum_{n=3}^{\infty }\frac{1}{n(\ln n)(\ln \ln n)^p}$$
$$f(x)=\frac{1}{x(\ln x)(\ln \ln x)^P}$$
$$\int_{3}^{\infty }\frac{1}{x(\ln x)(\ln \ln x)^p}$$
I stucked at her... | Another way is to use the Cauchy condensation test:
For a non-negative, non-decreasing sequence $a_n$ of reals, we have
$$\sum_{n=1}^\infty a_n<\infty\;\;\Leftrightarrow\;\;\sum_{n=0}^\infty 2^n a_{2^n}<\infty$$
Applied to your situation we get that your series converges if and only if
$$\sum_{n=2}^\infty \frac{1}{n (\... | {
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Finding dimension or basis of linear subspace Let $\;F\;$ be some field and take a column vector $\;\vec x=\begin{pmatrix}x_1\\x_2\\\ldots\\x_m\end{pmatrix}\in F^m\;$ . We defined
$$W=\left\{\;A\in V=M_{n\times m}(F)\;:\;\;A\vec x=\vec 0 \;\right\}$$
It is asked: is $\;W\;$ a linear subspace of $\;V\;$ and if it is fin... | Here's one way to think about it: If $Ax = 0$, then each of the rows of $A$ is orthogonal to $x$. The collection of vectors orthogonal to a nonzero vector is an $(m-1)$-dimensional subspace**. Thus to specify a matrix $A$ with $Ax = 0$, we must choose $n$ rows each from this $(m-1)$-dimensional space, which implies t... | {
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Prove that the complex expression is real Let $|z_1|=\dots=|z_n|=1$ on the complex plane.
Prove that:
$$
\left(1+\frac{z_2}{z_1}\right)
\left(1+\frac{z_3}{z_2}\right)
\dots
\left(1+\frac{z_n}{z_{n-1}}\right)
\left(1+\frac{z_1}{z_n}\right)
\in\mathbb{R}
$$
I have tried induction and writing every "subexpression" as $(1+... | Writing $z_k = e^{i\theta_k}$, $k = 1,2,\ldots, n$, we write the above expression as
\begin{align}&(1 + e^{i(\theta_2 - \theta_1)})(1 + e^{i(\theta_3 - \theta_2)})\cdots (1 + e^{i(\theta_n - \theta_{n-1})})(1 + e^{i(\theta_1 - \theta_n)})\\
&= e^{-i(\theta_2 - \theta_1)/2}e^{-i(\theta_3 - \theta_2)/2}\cdots e^{-i(\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1184048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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An example of a Ring with many zero divisors Is there an example of a commutative ring $R$ with identity such that all its elements distinct from $1$ are zero-divisors?
I know that in a finite ring all the elements are units or zero-divisors. Is there a finite ring with the property I've required?
Obviosuly I'm requiri... | Hint $\ $ If $\,1\,$ is the only unit then $\,-1 = 1\,$ so the ring is an algebra over $\,\Bbb F_2.\,$ With that in mind, it is now easy to construct many examples.
Such commutative rings - where every element $\ne 1$ is a zero-divisor - were called $0$-rings by Paul M. Cohn. Clearly they include Booolean rings, i.e. r... | {
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"source": "stackexchange",
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Simplify the expression $\sin 4x+\sin 8x+\cdots+\sin 4nx$
Simplify the expression $\sin 4x+\sin 8x+\cdots+\sin 4nx$
I have no idea how to do it.
| Alternatively,
$\sum_{k=1}^n \sin(4kx) = \Im\left\{\sum_{k=1}^n \exp(4kxi)\right\} = \Im\left\{\frac{1-\exp(4(n+1)xi)}{1-\exp(4xi)}\right\} = \Im\left\{\frac{(1-\exp(4(n+1)xi)\exp(-2xi))}{\exp(-2xi)-\exp(2xi))}\right\} = \frac{\cos(2x)(1-\cos(4(n+1)x))-\sin(2x)\sin(4(n+1)x)}{2\sin(2x)}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solving the system of differential equations I need to solve the system of equations
$$tx'=\begin{pmatrix}
2 & -1 \\
3 & -2 \\
\end{pmatrix}x$$
where $t>0$. I can solve this system if there was not any $t$ there. How do I treat that $t$?
Any help would be appreciated!
Thanks in advance!
| Change variable $\tau=\log(t)$
Then
$$
t dx/dt= dx/d(\log(t))=dx/d\tau
$$
Solve the system with respect to $\tau$ and replace finally $\tau$ by $\log(t)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $A \subseteq B$ and $B \subset C$, then $A \subset C$. Prove that if $A \subseteq B$ and $B \subset C$, then $A \subset C$.
Proof:
$A \subseteq B \Longrightarrow \forall x\in A, x \in B.$ Since $B \subset C$, it follows that $x \in B \Longrightarrow x \in C$ but $\exists c \in C \ni c \notin B.$
Since $A... | If A is a subset B then for all A is element of B. It follows B is a subset of C for all B is element of C. Since A is element of B for all A and B is element of C for all B Therefore A is subset of C.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1184525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If a group $G$ contains an element a having exactly two conjugates, then $G$ has a proper normal subgroup $N \ne \{e\}$ If a group $G$ contains an element a having exactly two conjugates, then $G$ has a proper normal subgroup $N \ne \{e\}$
So my take on this is as follows: If we take $C_G(S)$ of S. This is a subgroup o... | Simplifying $a,b$ argument a little:
Let $g$ be the element with exactly two conjugates. Suppose $C_G(g) = \{e\}$. Since $g \in C_G(g)$, this means $g = e$.
By Lagrange's Theorem, $[G : C_G(g)] = 2$ imples $\lvert G \rvert = 2$, so $G = \{e, h\}$. The two conjugates of $g$ are itself and $hgh^{-1}$, but both are $e$, ... | {
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Prove $(1 +\frac{ 1}{n}) ^ {n} \ge 2$ Using induction, I proved the base case and then proceeded to prove: $$(1 + \frac{1}{n+1}) ^ {n+1} \ge 2$$ given $$(1 + \frac{1}{n}) ^ n \ge 2$$ However, I'm stuck at this point and have no clue how to go about it. Other than induction, I tried simple algebraic transformations but ... | Assuming $\left ( 1+\dfrac{1}{n}\right)^n \geq 2$, we want to prove $\left ( 1+\dfrac{1}{n+1} \right)^{n+1}\geq 2$.
You may start by saying: $\left ( 1+\dfrac{1}{n+1}\right)^{n+1} = \left ( 1+\dfrac{1}{n+1} \right)^n\left ( 1+\dfrac{1}{n+1}\right).$
But if $\left( 1+\dfrac{1}{n}\right)^n\geq 2, $ then $\left( 1+\dfrac... | {
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"source": "stackexchange",
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"answer_id": 2
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Hartshorne's Algebraic geometry Chapter III ex. 9.10 I'm struggling with the exercise said in the title (Hartshorne, III, ex. 9.10). No problems in showing that $\mathbb{P}^1$ is rigid. In the second part, we want to show that $X_0$ being rigid does not imply that $X_0$ does not admit global deformations.
The problem ... | Consider a field of characteristic $\ne 2$ , and the projective subscheme $X \subset \mathbb A^2_k\times_k\mathbb P^2_k $ given by the equation $(a-1)x^2+(b-1)y^2+z^2=0$.
The projection morphism $f:X\to \mathbb A^2_k$ is smooth above $$S=D(a-1,b-1)=\operatorname {Spec}k[a,(a-1)^{-1},b,(b-1)^{-1}]\subset \mathbb A^2_k\... | {
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"source": "stackexchange",
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if G is generated by {a,b} and ab=ba, then prove G is Abelian if G is generated by {a,b} and ab=ba, then prove G is Abelian
All elements of G will be of the form $a^kb^j,\,\, k,j \in \mathbb{Z}^+$
so I need to get $a^kb^j = b^ja^k$
| First notice that $G=<a,b>$ does not imply directly that all elements of $G$ are of the form $a^kb^j$.
Here we have to different assertion the first one is $G=<a,b> \,\,\,(1)$ and the second one is $ab=ba\,\,\, (2)$. If this two assertions are true than the $G$ is Abelian. To understand the consequences of $(2)$ over $... | {
"language": "en",
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Evaluating the improper integral $\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$ I am supposed to solve this integral but i have no idea how:
$$\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$$
Since one limit is $0$ it will be divided by zero.
Can someone please explain this to me (I really want to understand) a... | Use the transformation $x=z^2$, to get $$I=6\int_{0}^1 \ln z dz $$ and then use integration by parts to obtain $I=-6$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Tricky 3d geometry problem We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get?
Astoundingly the answer is D. And here is a demonstration:
Well now I'm required to calculate the volume of this monster. ... | If we place the cube with its main diagonal from $(0,0,0)$ to $(1,1,1)$ and three edges along the axes, then we can parametrize two edges and a diagonal:
$$
\begin{align}
edge_1&:s\mapsto\begin{pmatrix}s\\0\\0\end{pmatrix},\quad s\in[0,1]\\
edge_2&:s\mapsto\begin{pmatrix}1\\0\\s\end{pmatrix},\quad s\in[0,1]\\
diag&:x\m... | {
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"source": "stackexchange",
"question_score": "15",
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"answer_id": 3
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How to explain the formula for the sum of a geometric series without calculus? How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that
$$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 - (\frac{1}{4... | What about multiplying the LHS by $(1 - \frac{1}{4})$?
Or is that what you wanted to avoid?
I mean it is not so difficult to understand that nearly all terms cancel... ?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 13,
"answer_id": 3
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simplifying an expression which concludes $e^x$ I am solving an high school question right now(well, I'm an high-scholar) and I'm don't understand how they simplify the expression:
$$\frac{-e^{x}+\frac{e^{2x}}{\sqrt{1+e^{2x}}}}{-e^{x}+\sqrt{1+e^{2x}}}$$
To be the expression $$\frac{-e^{x}}{\sqrt{e^{2x}+1}}.$$
I tried t... | Look at the numerator of your big fraction, which is $$-e^x+\frac {e^{2x}}{\sqrt{1+e^{2x}}}$$
Put this over a common denominator $$\frac {-e^x\sqrt{1+e^{2x}}+e^{2x}}{\sqrt{1+e^{2x}}}$$
Extract a factor $-e^x$ from the numerator
$$-e^x\frac {\sqrt{1+e^{2x}}-e^{x}}{\sqrt{1+e^{2x}}}$$
Now divide this by the original denom... | {
"language": "en",
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"source": "stackexchange",
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"answer_id": 2
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Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the plane $x + z = 1 $. Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the
plane $x + z = 1 $.
Hi all, i cant seem to get the correct answer for this question. The answer is $\pi$ but i got $... | Project the whole situation onto the $(x,z)$-plane, and draw a figure showing this plane. You will then realize that the body $B$ in question is half of the cylinder $$\bigl\{(x,y,z)\>|\>x^2+y^2\leq 1, \ 0\leq z\leq2\bigr\}\ .$$
It follows that ${\rm vol}(B)={1\over2}\cdot \pi\cdot 2=\pi$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Topology, Basis of a given topology. We defined basis for a topology, and there is something that I do not understand. Here is how we defined the basis.
Given a topological space $\left(X,\mathscr T\right)$
we defined basis for the topology to be the set $\mathscr B$
, consisting of subsets of $X$
if it satisfies ... | If we already have a topology given on a set, then a basis of the topological space must be a subset of the topology given on the set and it must generate ( i.e. by taking arbitrary unions, finite intersections of the basis elements) the whole topological space. So, if you choose any topology on a set and try to find a... | {
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Why is $I^1=\varphi*I^0$?
Why is $I^1=\varphi*I^0$ ?
where, $I^j(n)=n^j$ and $(f*g)(n)=\sum\limits_{d\in\mathbb N, d\mid n}f(d)g(\frac{n}d)$
$\varphi$ is the Euler-totient function, $\varphi(c)=\left(\prod\limits_{p\text{ prime},\atop p\mid c}\frac{p-1}{p}\right)c$
So $I^1(n)=n\overset!=\varphi*I^0(n)=\sum\limits_{d\... | The convolution $\varphi * I^0$ is commutative, so
$$\varphi * I^0 =\sum_{d\mid n}\varphi(n)\left(\frac{n}{d} \right)^0= \sum_{d\mid n} \varphi\left(\frac{n}{d}\right).$$
It is conventient to look at the latter term, rather than the second one, because it is easier to see what is going on.
The Euler totient function co... | {
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"source": "stackexchange",
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total variation of uniformly bounded function If a function is uniformly bounded and has finite variation, is the finite variation less than the uniform bound times a constant?
Thank you.
| No.
Take functions of $C([0;1])$ for example. Take $f_n(0)=0$ and $f_n'(x)=1$ if $\frac{2k}{n}\leq x < \frac{2k+1}{n}$ for some integer $k$ and $f_n'(x)=-1$ otherwise. those functions are saw shaped, you can easily show that $||f_n||_\infty = \frac{1}{n}$ and the total variation of $f_n$ is always one.
Reciprocally if ... | {
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How to get value of $\lim\limits_{n \to \infty} \frac{\sqrt[n]{n!}}{n}$ without using Stirling-formula? The problem $$\lim\limits_{n \to \infty}\frac{\sqrt[n]{n!}}{n}$$ was a for-fun-only exercise given by our Calculus Professor. I was able to solve it quite easily with the use of the Stirling-formula, but can't figure... | Start as
$$ \frac{(n!)^{1/n}}{n} = e^{ \frac{1}{n}\ln n! - \ln n } \sim e^{ \frac{n\ln n-n+1}{n}-\ln n }= e^{-1+1/n} \longrightarrow ... $$
| {
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"timestamp": "2023-03-29T00:00:00",
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Probability proof of inversion formula for Laplace transform Let $f:[0, \infty[\longrightarrow \mathbb{R}$ be bounded and continuous and define $L(\lambda)=\int_0^\infty e^{-\lambda x}f(x)dx$.
Let $X_n$ be a sequence of independent random variables with exponential distribution of rate $\lambda$.
Using the fact that t... | Old question, but I was working on the same problem recently. Here's an idea:
The functions $g_n$ defined by $g(x) = f(x/n)$ are bounded and continuous. Let $L_h$ denote the Laplace transform of a function $h$, and let the independent exponential random variables $X_1,X_2,\dots$ have rate $\frac{1}{y}$ (using $y\neq 0$... | {
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Negative modulus In the programming world, modulo operations involving negative numbers give different results in different programming languages and this seems to be the only thing that Wikipedia mentions in any of its articles relating to negative numbers and modular arithmetic. It is fairly clear that from a number ... | Negating the modulus preserves the congruence relation, by $\ m\mid x\color{#c00}\iff -m\mid x,\,$ so
$\quad a\equiv b\pmod m\iff m\mid a\!-\!b \color{#c00}{\iff} -m\mid a\!-\!b\iff a\equiv b\pmod {\!{-}m}$
Structurally, a congruence is uniquely determined by its kernel, i.e. the set of integers $\equiv 0.\,$ But this ... | {
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Polynomial subspace Wondering abut this set : $E=(p(X) \ \in \mathbb{R}[X]; Xp(X)+p'(X)=0)$, is it a subspace of $\mathbb{R}[X] $?
I definitely think it is because it only includes the zero polynomial but how could we prove it ? I usually take $u$ and $v$ which are in the set and then prove that $\lambda u+v \ \in$ the... | Hint: Find all elements of the set $E$.
Hint 2: Look at the degrees and leading coefficients.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 2
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$|G:H|=2$, $K \leq G$ with an element not in $H$. Prove $HK=G$ I know it has something to do with normal subgroups and cosets partitioning, but I don't know how to deal with cosets of $HK$.
| Since $H$ has index $2$, it is a normal subgroup. This implies that $HK$ is a subgroup. (In general it is a subset). But $H \subsetneq HK \subseteq G$, where the first inclusion is strict because $K$ has an element not in $H$. Comparing indices it follows that $G=HK$.
| {
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How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$ I need to show the following:
$$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$
For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
| Presumably you know that $\lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)^n = e$. Try setting $n = \frac{k}{z}$ where $z$ is a constant and see where that gets you.
| {
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Proof of aperiodic Markov Convergence Theorem for null recurrent case. Status quo:
We consider a irreducible, aperiodic Markov chain $(X_n)_{n\in\mathbb{N}}$ on a countable set $S$ with tranistion function $p(\cdot,\cdot)$.
Now we want to examine $\underset{n\rightarrow\infty}{\text{lim}}~p^n(i,j)$ for arbitrary $i,j... | After some more research I found a nice answer in the book by Norris Theorem 1.8.5. It also makes use of a Coupling argument with the choice of a shifted measure.
| {
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How to solve a congruence system? Here is a tricky congruence system to solve, I have tried to use the Chinese Remainder Theorem without success so far.
$$2x \equiv3\;(mod\;7)\\
x\equiv8\;(mod\;15)$$
Thank you very much
Li
| $$ 2x = 3(mod 7) $$
$$ x+x= 3(mod 7) $$
$$ x = 5(mod 7) $$
$$ x = 8(mod 15) $$
$$ x = 15k_1 + 8 $$
$$ x = 7k_2 + 5 $$
$$ x = 105k_3 + c $$
$$ 0 \le c < 7*15 $$
$$ c(mod 7)=5 $$
$$ c = 15k_4+8 $$
$$ 0 \le k_4 < 7 $$
$$(15k_4+8)(mod 7)=3$$
$$ k_4 + 1 = 5 $$
$$ k_4 = 4 $... | {
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primitive roots problem. that integer n can never have exactly 26 primitive roots. Show that no integer $n$ can have exactly 26 primitive roots.
I know that if $n$ has primitive roots then it has exactly $\phi(\phi(n))$ primitive roots.
I think the proof has to use contradiction.
Suppose $n$ is an integer and has exa... | Because 26 is what is called a "nontotient." With only two exceptions, $\phi(n)$ is never odd. There are also some even values that never occur as $\phi(n)$, and 26 is one of them.
Remember that $\phi(p) = p - 1$ if $p$ is prime. But $27 = 3^3$, so 26 is not a totient this way. How about $\phi(p^\alpha) = (p - 1)p^{\al... | {
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What multiples of $d$ are still multiples of $d$ when they have their digits reversed? I teach at a school for 11 to 18 year olds. Every term I put up a Challenge on the wall outside my classroom.
This question is one that I have devised for that audience. I think that it is quite interesting and I would like to share ... | I think this question is fascinating, thanks for posting it. In addition to $3$ and $9$, I might add that multiples of $11$ are also always reversible. Therefore, multiplies of $33$ and $99$ are also always reversible. One might hope that squaring an always reversible number leads to an always reversible number, but... | {
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Closure of Intersection Given a topological space $\Omega$.
By minimality it holds:
$$\overline{A\cap B}\subseteq\overline{A}\cap\overline{B}$$
Especially one has equality for:
$$A=\overline{A}:\quad\overline{A\cap B}=A\cap\overline{B}$$
How to check this quickly?
| What if $A= [0,1]$ and $B=(1,2),$ then $A \cap B = \emptyset$ so $\overline{A \cap B} = \emptyset,$ but $A \cap \overline{B}= [0,1] \cap [1,2]=\{1\}.$ So you don't have the equality.
| {
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Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$ that is closest to the point $(3, 0, 0)$ Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$
that is closest to the point $(3, 0, 0)$
Hi all, could someone give me a hint on how to start doing the above question?
| Set $P=(3,0,0)$.
The function of interest:
$$f(x,y)=\left(\frac{x}{2}\right)^2+\left(\frac{y}{5}\right)^2$$
Its graph is given by:
$$\phi (u,v)=(u,v,f(u,v))$$
We calculate the normal-vector as follows:
$$\frac{\partial \phi }{\partial u}\times \frac{\partial \phi }{\partial v}(u,v)=\left(-\frac{u}{2},-\frac{1}{25} (2 v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Method of finding a p-adic expansion to a rational number Could someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$
| The short answer is, long division.
Say you want to find the $5$-adic expansion of $1/17$. You start by writing
$$
\frac{1}{17}=k+5q
$$
with $k \in \{0,1,2,3,4\}$ and $q$ a $5$-adic integer (that is, a rational number with no powers of $5$ in the denominator). Then $k$ is the first term in the expansion, and you repeat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Boolean Algebra: Simplifying $\;xyz + x'y + xyz'$ Given the following expression: $xyz + x'y + xyz'\,$ where ($'$) means complement, I tried to simplify it by first factoring out a y so I would get $\;y(xz + x' + xz').\,$
At this point, it appears I have several options:
A) Use two successive rounds of distributive pr... | Using the distributive property (first method), we get:
$$\begin{align} xyz + x'y + xyz' & = xy(z + z') + x'y \\ &= xy + x'y \\&= (x + x')y \\&= y\end{align}$$
You erred when you went from $ y ( z + x' + z') $ to $yx'$. You should have $$y((z+z')+x')= y(1+x') = y\cdot 1 = y$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Laplace Transform for a difficult function The Laplace Transform I'm having trouble with is:
$$f(t) = 6te^{-9t}\sin(6t)$$
I'm not sure what the protocol is for multiplying t into it.
The Laplace Transform for $f(t) = 6e^{-9t}\sin(6t)$ is $\dfrac 6{(s+9)^2 - 36}$.
Can't figure out how to add in the $t$.
Thanks in advanc... | Here is a partial solution:
\begin{align}
\mathcal{L}\{f(t)\}&=\mathcal{L}\{e^{-9t}[6t\sin(6t)]\} \\
&=\mathcal{L}\{6t \sin(6t)\} \vert_{s \to s+9} & \text{First Translation Theorem}\\
&= 6 \mathcal{L}\{t\sin(6t)\} \vert_{s \to s+9} & \text{linearity} \\
&=-6\frac d{ds} \mathcal{L}\{\sin(6t)\} \vert_{s \to s+9} & \text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How many ways are there to divide elements into equal unlabelled sets? How many ways are there to divide N elements into K sets where each set has exactly N/K elements.
For the example of 6 elements into 3 sets each with 2 elements. I started by selecting the elements that would go in the first set (6 choose 2) and the... | Hint: You are close. As the sets are unlabeled, choosing $\{a,b\},\{c,d\},\{e,f\}$ is the same as choosing $\{e,f\},\{c,d\},\{a,b\}$, but you have counted them both.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Gromov's boundary at infinity, drop the hypothesis on hyperbolicity It's an easy result that if we have two quasi isometric hyperbolic spaces, then their Gromov boundaries at infinity are homeomorphic.
I found online these notes where at page 8, prop 2.20 they seem to drop the hypothesis on hyperbolicity. They give two... | Counter-examples are due to Buyalo in his paper "Geodesics in Hadamard spaces" and to Croke and Kleiner in an unpublished preprint. See also the later paper of Croke and Kleiner entitled "Spaces with nonpositive curvature and their ideal boundaries" which cites their preprint and gives further information.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Computing Residue for a General, Multiple-Poled function? I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant:
Compute $\,Res_f(a)$ for the following function:
$$f(z) = \frac{1}{(z - a)^3}\ \tanh z $$
What I'm confused about is the ambiguit... | You need to separate between a few cases:
$a=i(\frac{\pi}{2}+\pi k)$
In this case we have $\lim_{z \to a}(z-a)^4f(z)$ is finite and different from $0$ so $a$ is a pole of order $4$
$a=\pi ki$
In this case $\lim_{z \to a}(z-a)^2f(z)$ is finite and different from $0$ so $a$ is a pole of order $2$
$a \ne i(\frac{\pi}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question about one-to-one correspondence between left coset and image of Homomorphism Let $G$ be a finite group and $K$ be a homomorphism such that $K : G\rightarrow G^*$ and $H=\ker K$.
Then there is a one-to-one correspondence between the number of left cosets of $\ker K$ and the number of elements in $K[G]$.
But, it... | The Emperor of Ice Creams' answer is letter-perfect from a mathematical point of view. Since my comment is rather long, I'm writing it is an answer (but it's only an "illustration").
Suppose $G = D_4 = \langle r,s: r^4 = s^2 = e, sr = r^3s\rangle$ (this is the dihedral group of order 8), and that $G' = \langle a,b: a^2... | {
"language": "en",
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How do I define a block of a $(0,1)$-matrix as one that has no proper sub-blocks? I'm struggling to come up with a definition of a "block" in a $(0,1)$-matrix $M$ such that we can decompose $M$ into blocks, but the blocks themselves don't further decompose. This is what I've got so far:
Given any $r \times s$ $(0,1)$... | I would suggest breaking this up into two definitions... maybe call one of them a "block", and another a "simple block". That is:
Given any $r \times s$ $(0,1)$-matrix $M$, we define a block of $M$ to be a submatrix $H$ in which: (a) every row and every column of $H$ contains a $1$, (b) in $M$, there are no $1$'s in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Need clarification on a Taylor polynomial question $$f(x) = 5 \ln(x)-x$$
second Taylor polynomial centered around $b=1$ is $-1 + 4(x-1) - (5/2)(x-1)^2$
let $a$ be a real number : $0 < a < 1$
let $J$ be closed interval $[1-a, 1+a]$
find upper bound for the error $|f(x)-T_2(x)|$ on interval $J$
answer in terms of a
so i ... | We are given:
$$f(x) = 5 \ln(x) - x$$
The second Taylor polynomial centered around $b=1$ is given by:
$$T_2(x) = -\frac{1}{2} 5 (x-1)^2+4 (x-1)-1$$
We are told to let $a$ be a real number such that $0 \lt a \lt 1$ and let $J$ be the closed interval $[1 −a, 1 +a]$.
We are then asked to use the Quadratic Approximation Er... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative tests question Show that $k(x) = \sin^{-1}(x)$ has $0$ inflections $2$ critical points $0$ max/min
I find that the first derivative is $$\frac{1}{\sqrt{1-x^2}}$$
Second derivative is $$\frac{x}{(1-x^2)\sqrt{1-x^2}}$$
I don't know why it has $2$ critical points, I can find only one at second derivative which ... |
In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined
As you can easily see, $f'(x)$ is undefined on $x=1$ and $x=-1$.
I think you might have misunderstood the concept of a critical point for th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1187991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Equation of plane parallel to a vector and containing two given points
I'm not sure how to solve this.
I started by finding the equation of the line AB.
| Hints:
Let the wanted plane be $\;m: ax+by+cz+d=0\;$ , then it must fulfill:
$$\begin{align}&0=(a,b,c)\cdot(1,-5,-3)\implies&a-5b-3c=0\\{}\\
&A\in m\implies&3a-2b+4c+d=0\\{}\\
&B\in m\implies&2a-b+7c+d=0\end{align}$$
Well, now solve the above linear system. Pay attention to the fact that you shall get a parametrized an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Series solution to first order differential equation I need to find a series solution to the following simple differential equation
$$x^2y'=y$$
Assuming the solution to be of the form $y=\sum a_nx^n$ and equating the coefficients on both the sides, all the coefficients turn out to be zero which is definitely wrong.
Pl... | Well,
\begin{eqnarray}
\int \frac{dy}{y} &=& \int \frac{dx}{x^{2}}\\
\implies \ln y &=& -\frac{1}{x}+c
\end{eqnarray}
Hence,
\begin{equation}
y=Ae^{-\frac{1}{x}}
\end{equation}
Is a series solution necessary?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Connection between algebraic geometry and complex analysis? I've studied some complex analysis and basics in algebraic geometry (let's say over $\mathbb C $). We had been mentioned GAGA but nothing in detail. Anyway, from my beginner's point of view, I already see many parallels between both areas of study like
*
*c... | May I recommend Mme. Raynaud: Géométrie Algébrique et Géometrie Analytique, Exposé XII. The GAGA-relation is made precise by a functor
$an: \underline{Al}_\mathbb{C} \xrightarrow{} \underline{An},\ X \mapsto X^{an},$
from the category of schemes over $\mathbb{C}$ which are locally of finite type and the category of c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
random variables independence I need to check if Z and W are dependent or not.
X,Y ~ Exp(2)
Then I define: Z=X-Y , W=X+Y.
Now How I can check that Z and W are dependent or not ?
I know from the teory that I should show that f(z,w) = f(z)f(w) , but I dont see how I do it here.
Thanks for help.
| If $Z$ and $W$ are indeed independent then e.g. $P(Z>1,W<1)=P(Z>1)P(W<1)>0$.
This contradicts that $P(Z<W)=1$ as mentioned in the comment of @Did.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.
To make such a mobius transformation i tried to send 3 points on the edge to 3 poin... | If $|z|=1$, then $\overline z=1/z$. So,
$$\left|{z-a\over 1-\overline az}\right|=\left|{z-a\over1-\overline{a/z}}\right|=\left|{\overline z(z-a)\over\overline z-\overline a}\right|=|\overline z|\left|{z-a\over\overline{z-a}}\right|=|\overline z|=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding integer part of a polynomial root If a polynomial $f(x)=x^3+2x^2+x+5$ has only one real root $\omega$ then integer part of $\omega = -3$.
It appeared in my school exam and I couldn't solve it completely but somehow tried to find the range in which the root lies. But it was tough to show that $[\omega] = -3$.
| $f'(x)=3x^2+4x+1$
The derivative vanishes for $x=-1,-\frac{1}{3}$ It is positive for $x\in(-\infty,-1)$
$f(-3)=-7$
$f(-2)=13$
Thus, by the intermediate value theorem there must be a root in $(-3,2)$ such that $f(\omega)=0$. Thus, it has an integer part of $-3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $y = x^N + z^N$ for N? Is there a way to solve equations of the form
$y = x^N+z^N$ for N?
I've been looking and I don't think there is a way, but I'm not sure if there's some obscure algebra rule that will help me here.
| As already said, there is no closed form for the solution (except for very specific cases) and only numerical methods would solve the problem.
Let us admit that $x,y,z$ are positive numbers and $N$ be a real number and consider Newton method which, starting from a reasonable guess $N_0$, will update it according to $$N... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Related rates problem. Determine the speed of the plane given the following information.
I started by noting that I am looking for $\displaystyle\frac{dx}{dt}$ and I am given $\displaystyle\frac{d\theta}{dt}=1.5^{\circ}/\mbox{s}$.
I then related $\theta$ to $x$ by $\displaystyle\cot(\theta)=\frac{x}{6000\mbox{m}}$, th... | you need to use radians, so you need to convert your $\frac{d\, \theta}{dt} = 1.5^\circ = \frac{1.5 \pi}{180}. $ putting that in $$\frac{dx}{dt} = -6000 \csc^2(\pi/3) \frac{1.5 \pi}{180} = -6000 \times \frac{4}{3} \times \frac{1.5 \pi}{180} = -209.44\ m/sec $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers An exam for high school students had the following problem:
Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on... | Let the circle be the unit circle. Consider the square of equal perimeter: it has side length $\pi/2$. Align this square so that the midpoint of its left side touches the leftmost point of the unit circle. Where is the upper right corner of the square? It has location $(-1+\pi/2,\pi/4)$, which is at distance $\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1188845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "251",
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Prove 3 is prime in the ring $\Bbb{Z[i]}$. I am not sure what is the right phrase for primes in some ring. The definition I was gave is that in a ring $A$, $p\in A$ is prime if $\forall x,y \in A$ such that $p\mid xy$, $p\mid x$ or $p\mid y$.
I don't know how to start.(I arrived at odd expressions but too long ones) a... | Suppose that $3=rs$. Then $9=N3=N(rs)=NrNs$. It follows then that say $Nr$ (up to sign) must be $1,3$ or $9$. I claim it cannot be $3$. Note that in such case, we can write $3=a^2+b^2$ for some integers $a,b$. But the squares modulo $4$ are $0,1$; and they add up to $0,1,2$, and $3$ is equivalent to none of $0,1,2$ mod... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is the notation $f=f(x)$ mathematically correct? Looking at the equation, which is written all over my textbooks, $f=f(x)$ or $\mathbf r = \mathbf r(x,y,z)$ or what-have-you, I can't help but think that that is just wrong. On the LHS is a function. On the RHS is the value of that function evaluated at some arbitr... | This notation tells you that the function will be called by two different names in the sequel. When the argument is important, it will be called $f(x)$. However when the argument is not particularly important, it will also be called $f$.
In this expression, the symbol $=$ is not used mathematically, but meta-mathemat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189012",
"timestamp": "2023-03-29T00:00:00",
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Proof or counterexample : Supremum and infimum If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum.
How to Prove This.
| The statement is false.
I bring you a counter example.
consider $\mathbb{Q}$ as the universal set with the following set definitions:
$$A_1=\{x\in \mathbb{Q}|x^2<2\}\cup{\{5\}}$$
$$A_2=\{x\in \mathbb{Q}|x^2<2\}\cup{\{6\}}$$
$$A_n=\{x\in \mathbb{Q}|x^2<2\}\cup{\{4+n\}}$$
In this example, sumpermum of $A_1$ is 5 and for ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Hausdorff measure of $f(A)$ where $f$ is a Holder continuous function. Let $f\colon \mathbb R^d\to \mathbb R^k$ be a $\beta-$ Holder continuous function ($\beta \in (0,1)$) and $A\subset \mathbb R^d$. As for a Lipschitz function $g$ it holds that $H^s(g(A))\leq Lip(g)^s H^s(A)$, also for $f$ should hold a similar ineq... | Your argument is correct, and there is no way to improve the estimate. A simple example can be given in the context of abstract metric spaces: let $X $ be the metric space $ (A,d_\beta)$ where $d_\beta(x,y)=|x-y|^\beta$ is a metric. Let $f$ be the identity map from $A$ to $X$; it has the property $|f(x)-f(y)| = |x-y|^... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there a Turing Machine that can distinguish the Halting problem among others? Can there be a Turing machine, that given two oracles, if one of them is the Halting problem, then this machine can output the Halting problem itself?
Clearly, if the first oracle is always the Halting problem, then such machine exists, j... | No. Using programs where both oracles give the same output, you obviously cannot distinguish them. Therefore the only way to distinguish the oracles is to analyse programs on which both oracles give different results.
However, if the two oracles give different results, the only way to decide which oracle is the one ans... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the derivative using the chain rule and the quotient rule
$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$.
Here is my work:
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4x^4\left(x+1\right)^3}{\left(x+1\right)^8}$$
I know the fin... | Not wrong, only written in a different form: you can simplify much of it.
It's much better exploiting the chain rule: if you call
$$
g(x)=\frac{x}{x+1},
$$
then $f(x)=(g(x))^4$ and so
$$
f'(x)=4(g(x))^3g'(x)=
4\left(\frac{x}{x+1}\right)^{\!3}\frac{1(x+1)-x\cdot 1}{(x+1)^2}=
\frac{4x^3}{(x+1)^5}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Compact subset of locally compact $\sigma$-compact Let $X$ be a non-compact, locally compact space. Also suppose that there is a sequence of compact non-empty sets $\{K_n\}_{n\in N}$ such that $$X=\bigcup_{n\in N} K_n,\quad K_n\subset K_{n+1}.$$
Now could we say that for any compact subset $K\subset X$ there is a numb... | No, not as stated.
Let $X = [0,1)$ be the half-open unit interval, which is locally compact but not compact. Let $K_n = \{0\} \cup [\frac{1}{n}, 1-\frac{1}{n}]$ which is compact. Then we have $K_n \subset K_{n+1}$ and $X = \bigcup_n K_n$. But the compact set $K = [0, \frac{1}{2}]$ is not contained in any of the $K_n$... | {
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"url": "https://math.stackexchange.com/questions/1189563",
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For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that $-5 \le x+y \le -2.$ I have a problem:
For $x, y \in \Bbb R$ such that $(x+y+1)^2+5x+7y+10+y^2=0$. Show that
$$-5 \le x+y \le -2.$$
I have tried:
I write $(x+y+1)^2+5x+7y+10+y^2=(x+y)^2+7(x+y)+(y+1)^2+10=0.$
Now I'm stuck :(
Any help will be a... | Since $(y+1)^2\ge 0$ we must have $(x+y)^2+7(x+y)+10=(x+y+5)(x+y+2)\leq 0$ then, solving this inequality for $x+y$, we get $$-5\le x+y \le -2$$
| {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
} |
How to do multiplication in $GF(2^8)$? I am taking an Internet Security Class and we received some practice problems and answers, but I do not know how to do these problems , an explanation would be greatly appreciated
Try to compute the following value:(the number is in hexadecimal and each represents a polynomial in ... | To carry out the operation, we need to know the irreducible polynomial that is being used in this representation. By reverse-engineering the answer, I can see that the irreducible polynomial must be $x^8+x^4+x^3+x+1$ (Rijndael's finite field). To carry out a product of any two polynomials then, what you want to do is m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Understanding controllability indices I'm teaching myself linear control systems through various online materials and the book Linear Systems Theory and Design by Chen. I'm trying to understand controllability indices. Chen says that looking at the controllability matrix $C$ as follows
$$ C = \begin{bmatrix} B & AB & A... | I am studying the same subject now, from the same book and had exactly the same question you had.
The answer by @Pait doesn't show that the resulting set $\mathit{\lambda_iAb_i}$ is a subset of the LHS vectors relative to $\mathit{A^2b_1}$ in the controllability matrix. However I had understood it myself, and I will po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1189957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Given a drawing of a parabola is there any geometric construction one can make to find its focus? This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci?
I think this is harder, I can't even find is axis or vertex... | As @AchilleHui mentions, the midpoints of two parallel chords lead to the point of tangency ($T$) with a third parallel line. Note that the line of midpoints is parallel to the axis of the parabola. By the reflection property of conics, the line of midpoints and the line $\overleftrightarrow{TF}$ make congruent angles ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Does $\{(1, -1),(2, 1)\}$ spans $\mathbb{R}^2$? please correct me Can anyone please correct me? my problem is in the proof part below
Q: Does $\{(1, -1),(2, 1)\}$ spans $\mathbb{R}^2$?
A:
$$c_1(1, -1) + c_2(2, 1) = (x, y)$$
$$c_1 + 2c_2 = x$$
$$-c_1 + c_2 = y$$
$$c_1 = x - 2c_2$$
$$-(x - 2c_2) + c_2 = y$$
$$-x + 2c_2 ... | Perhaps easier:
$$\begin{cases}&\;\;\;c_1+2c_2=x\\{}\\&-c_1+c_2\;\;=y\end{cases}\stackrel{\text{sum both eq's}}\implies3c_2=x+y\implies c_2=\frac{x+y}3$$
and substituting back, say in equation two:
$$c_1=\frac{x-2y}3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Solve the initial value problem $u_{xx}+2u_{xy}-3u_{yy}=0,\ u(x,0)=\sin{x},\ u_{y}(x,0)=x$
Solve the partial differential equation $$u_{xx}+2u_{xy}-3u_{yy}=0$$ subjet to the initial conditions $u(x,0)=\sin{x}$, $u_{y}(x,0)=x$.
What I have done
$$
3\left(\frac{dx}{dy}\right)^2+2\frac{dx}{dy}-1=0
$$
implies
$$\frac{dx... | We use Laplace transform method and free CAS Maxima
http://maxima.sourceforge.net/
Answer:
$$u=\frac{\sin(x+y)}{4}+\frac{y^2}{3}+xy+\frac34\sin\left(x-\frac{y}{3}\right)$$
2 method
*
*$D_x^2+2D_xD_y-3D_x^2=(D_x-D_y)(D_x+3D_y)$
*General solution of $u_x-u_y=0$ is $u_1=f(x+y)$
*General solution of $u_x+3u_y=0$ is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Can someone provide a simple example of the "pre-image theorem" in differential geometry? I only have a background in engineering calculus. A problem I am currently working on relates to something called a "pre-image theorem"
The theorem roughly states:
Let $f: N \to R^{m}$ be a $C^{\infty} $ map on a manifold N of
... | A few examples might make things clear:
(1) Let $n=2$ and $m=1$. Define $f:\mathbb{R}^2 \to \mathbb{R}$ by $f(x,y) = x^2 + y^2$. Then $S = f^{-1}(c) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 =c\}$, which is a circle with radius $\sqrt{c}$. We have $n-m=1$, and a circle is indeed a one-dimensional manifold.
(2) Let $n=3$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Radius of convergence of a power series whose coeffecients are "discontinuous" I have a power series:
$s(x)=\sum_0^\infty a_n x^n$
with
$a_n=
\begin{cases}
1, & \text{if $n$ is a square number} \\
0, & \text{otherwise}
\end{cases}$
What is the radius of convergence of this series?
I tried to use ratio test, but as$\li... | Although $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ and $\lim_{n\to\infty}\sqrt[n]{a_n}$ don't exist, you can still find $\limsup_{n\to\infty} \sqrt[n]{a_n}$. Since
$$
\sup_{m\ge n} \sqrt[m]{a_m}=1
$$
for all $n\in\mathbb{N}$, $\limsup_{n\to\infty}\sqrt[n]{a_n}=1$ and so the radius of convergence is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Stirling number of the second kind and combinations The Stirling number of the second kind, denoted by $S(n,r)$, is defined as the number of $r$-partitions on a set of $n$ elements. Let $\binom{n}{r}$ (which is read $n$ choose $r$) denotes the number of $r$-element subsets of an $n$-element set. How to show that
$$
\b... | We know that,
${n \choose r} = {n \choose {n-r}}$,
So it's sufficient to prove that,$${n \choose {n-r}} \le S(n,r)$$.
proof
For any $(n-r)$-subset of $n$ element set we can create (at-least one) $r$-partitions in the following way,
1) Put all the $(n-r)$ elements in a single partition.
2) partition the remaining(if any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What initial value do I have to take at the beginning? In my lecture notes there is the following example on which we have applied the method of characteristics:
$$u_t+2xu_x=x+u, x \in \mathbb{R}, t>0 \\ u(x,0)=1+x^2, x \in \mathbb{R}$$
$$$$
$$(x(0), t(0))=(x_0,0)$$
We will find a curve $(x(s), t(s)), s \in \mathbb... | Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$
$\dfrac{dy}{dt}=x+y=t+y$ , we have $y=y_0e^t-t-1=y_0e^x-x-1$
$\dfrac{du}{dt}=0$ , letting $u(0)=F(y_0)$ , we have $u(x,y)=F(y_0)=F((x+y+1)e^{-x})$
$u(x,1-x)=f(x)$ :
$F(2e^{-x})=f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Problem in understanding models of hyperbolic geometry I recently started reading The Princeton Companion to Mathematics. I am currently stuck in the introduction to hyperbolic geometry and have some doubts about its models.
Isn't the hyperbolic space produced by rotating a hyperbola? That is, isn't hyperbolic geometry... | It looks like you are talking about the Poincaré half-plane model of hyperbolic geometry (see https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model )
Be aware this is only a model of hyperbolic geometry.
For doing "real-life" hyperbolic geometry you need a "surface with a constant negative curvature " or pseudo ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Union/intersection over index families of intervals I am currently reading "Introduction to Topology" by Bert Mendelson (third edition).
At Chapter 1, Section 4 - Indexed Families of Sets. For Exercise 5, he asks the following:
Let $I$ be the set of real numbers that are greater than $0$.
For each $x \in I$, let $A_x... | There is no smallest member of $I$. Given any $x \in I$, you'll eventually come across $[0, \frac{x}{2}]$ in your intersection. The only common point to all those intervals is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Let $R=\{(x,y): x=y^2\}$ be a relation defined in $\mathbb{Z}$. Is it reflexive, symmetric, transitive or antisymmetric? Let $R=\{(x,y): x=y^2\}$ be a relation defined in $\mathbb{Z}$. Is it reflexive, symmetric, transitive or antisymmetric).
I'm having most trouble determining if this relation is symmetric, how can I ... | *
*The relation is not reflexive: $(2,2)\notin R$
*The relation is not symmetric: $(4,2)\in R$, but $(2,4)\notin R$
*The relation is not transitive: $(16,4)\in R$, $(4,2)\in R$, but $(16,2)\notin R$
Now, let's see if the relation is antisymmetric. Suppose $(x,y)\in R$ and $(y,x)\in R$. Then $x=y^2$ and $y=x^2$, whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I show logically equivalence without a truth table Show that $(p \rightarrow q) \wedge (p \rightarrow r)$ and $p \rightarrow (q \wedge r)$ are logically equivalent.
I tried to do this making a truth table but I think my teacher wants me to solve it using the different laws of Logical Equivalences.
Can anyone he... | Here is an approach
$$(p \to q) \wedge (p \to r) \equiv (\neg p \vee q) \wedge (\neg p \vee r) \equiv \neg p \vee (q \wedge r) \equiv p \to (q \wedge r)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Irrationality of $\sqrt[n]2$ I know how to prove the result for $n=2$ by contradiction, but does anyone know a proof for general integers $n$ ?
Thank you for your answers.
Marcus
| Suppose that $\sqrt[n]2$ is rational. Then, for some $p,q\in\mathbb Q$,
$$\sqrt[n]2=\frac{p}{q}\implies 2=\frac{p^n}{q^n}\implies p^n=2q^n=q^n+q^n.$$
Contradiction with Fermat last theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Cohomology of Segre varieties Let $\Sigma_{n,m}$ be a Segre variety, i.e. the image of the Segre map $\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^{(n+1)(m+1)-1}$.
Then how can I calculate the first cohomology group of its tangent bundle, i.e. $H^1(\Sigma_{n,m},\mathcal{T})$?
| Forget about Segre and use the Künneth formula for $T=T_{\mathbb P^n\times {\mathbb P^m}}=T_{\mathbb P^n}\boxtimes T_{\mathbb P^m}$, obtaining:$$ H^1(\mathbb{P}^n\times\mathbb{P}^m,T)\\=[ H^0(\mathbb P^n,T_{\mathbb P^n})\otimes H^1(\mathbb{P}^m,T_{\mathbb P^m})]\oplus [H^1(\mathbb P^n,T_{\mathbb P^n}) \otimes H^0((\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I change the order of integration of this integral? $$\int\limits_1^e\int\limits_{\frac{\pi}{2}}^{\log \,x} - \sin\,y\, dy\,dx$$
I don't understand how to change the order because of the $\log\,x$ as the upper bound for the inner integral
How do I change it so it looks like $$\int \int f(x)d(y)\,dx\,dy$$
Thanks
| Your region is $S=\{(x,y): 1\leq x\leq e, \log x\leq y\leq \pi/2 \}$. You split it in two regions, $S_1=\{(x,y):1\leq x\leq e, 1\leq y\leq \pi/2\}$ (a rectangle) and $S_2=\{(x,y):0\leq y\leq 1, 1\leq x\leq e^y\}$. Then you write your integral as a sum over each of these regions, and here you have the integration in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximal Function Estimate Suppose $\psi$ is a rapidly decreasing function; i.e. for all $N>0$ there exists a constant $C_{N}$ such that $\left|\psi(x)\right|\leq C_{N}(1+\left|x\right|)^{-N}$. Define a family of functions $\left\{\psi_{j,k}\right\}_{j,k\in\mathbb{Z}}$ by $\psi_{j,k}(x)=2^{j/2}\psi(2^{j}x-k)$. For each ... | So this is not as nice as I would have liked. I'm hoping somebody will come up with something cleaner.
Set $\delta:=(1+\left|2^{j}x-k\right|)$ for $j,k\in\mathbb{Z}$. Without loss of generality, assume $f\geq 0$. Observe that
$$\left|a_{jk}\right|\leq\int_{\left|2^{j}y-k\right|\leq\delta}f(y)\left|\psi_{jk}(y)\right|dy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Probability and elementary set theory proofs (i) suppose P(A) = $0 $ or $ 1$, prove that A and any subset B of Ω are independent.
I did: $P(A)+P(B)+P(C)=1 $ since $ P(Ω)=1.$
If $P(A)=0, $ then $P(B)=1-P(C)=P(B|A)$
Similarly, if $P(A)=1, $ then $P(B)=P(C)=P(B|A)$
Therefore, A and any subset B of Ω are independent. Is ... | This is a bad assumption:
$$P(A)+P(B)+P(C)=1.$$
In the case $P(A) = 1,$ you cannot use this to prove something about
"any subset $B$ of $\Omega$", because the assumption implies that $P(B) = 0$.
We know there are subsets of $\Omega$ that have non-zero probabilities,
and you have discarded all of them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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