Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve the differential equation 2y'=yx/(x^2 + 1) - 2x/y I have to solve the equation:
$$ 2y'= {\frac{xy}{x^2+1}} - {\frac{2x}{y}} $$
I know the first step is to divide by y, which gives the following equation:
$$ {\frac{2y'}{y}} ={\frac{x}{x^2+1}} - {\frac{2x}{y^2}}$$
According to my notes I get that I should make a su... | Hint
$$2y'= {\frac{xy}{x^2+1}} - {\frac{2x}{y}}$$
You can also multiply by y the Bernouilli 's equation
$$2y'y= {\frac{xy^2}{x^2+1}} - 2x$$
Observe that $(y^2)'=2y'y$ substitute $z=y^2$
$$z'= {\frac{xz}{x^2+1}} - 2x$$
Now it's a linear first ode
But you can do it your way
$${\frac{2y'}{y}} ={\frac{x}{x^2+1}} - {\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solution to a-concentric-circle-problem This question has been solved via trigonometry. I am trying to solve it geometrically.
Referring to the figure, I translate DE to BB’ (through //gms $DEE’D’$ and $BE’D’B’$). Also, I rotate $AB$ to $AA’$. As a result, the condition $DE + BC = AB + AC$ is equivalent to saying $\tri... | I asked this question, it has been solved via trigonometry as you mentioned but some days ago I solve it with a shorter solution because it is too easy I just summarize it :
it is clear if $\angle A$$=90$ we are done therefor at first suppose $\angle A$$>90$ obviously $BE$>$AB$ and $DC$>$AC$ so $DE$>$AB$+$AC$-$BC$ , ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Is there a way to prove algebraically that a Möbius strip is non-orientable? I am doing my HL Maths coursework on non-orientability of surfaces and am trying to prove whether a möbius strip is orientable or not (of course it isn't)
Is there a way to prove algebraically that a mobious strip is non-orientable via vectors... | Essentially, you just need a parity argument. A Moebius strip can be seen as a quotient of a square in $\mathbb{R}^2$, where the "left" and "right" sides are identified but glued together by firstly reversing the orientation of one of them. To help you visualizing this, in the following diagram points labelled by the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$x^3+2x+2 \in \mathbb{F}_3[x]$. Let $\alpha$ be a root in some extension field. One can see by brute force that $x^3+2x+2$ has no roots in GF(3). So it is irreducible and hence the minimal polynomial of $\alpha$.
My question is what is $\alpha$, how can I think about it? I determined that $GF(3)(\alpha) \cong GF(3^3)=G... | Careful, $\operatorname{GF}(27) \neq \mathbb Z/27\mathbb Z$ !
The latter is a ring of order $27$ which has zero-divisors ($3 \cdot 9 = 0$) and so not a field, and whose elements are $0$ up to $26$.
The first is a field of order $27$ of characteristic $3$, in which $\{0,1,\ldots, 26\} = \{0, 1, 2\}$ because $4=1$ etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
If every subsequence of $(x_n)$ has a subsequence converging weakly to $x$ then $x_n$ converges weakly to $x$. Let $H$ be a Hilbert space(or a reflexive Banach space) and $(x_n)$ a sequence in $H$. Is the following proposition true?
If every subsequence of $(x_n)$ has a subsequence converging weakly to $x$ then $x_n$ c... | This is in general true for any sequence in a topological space.
Let $X$ be a topological space, $\left\{x_n\right\}\subset X$ a sequence, and $x\in X$. Then the following are equivalent:
*
*$x_n$ converges to $x$;
*any subsequence of $\left\{x_n\right\}$ admits a subsequence which converges to $x$.
Proof: $(1)\Rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Probability and combinatorics problems - picking balls and choosing postcards I am sorry for bothering you with such a trivial and easy question (in comparison to the others asked here) but I have no idea where else I could ask. These are two problems I have to solve and I just need you to check if my solution is corre... | Your method for the first problem is the probability of selecting a red, followed by $3$ blues, sequentially. We have to take into account all the different ways to arrange $3$ blues and $1$ red. Note that this is a hypergeometric random variable, giving probability
$$\frac{{10 \choose 1}{3 \choose 3}}{{13 \choose 4}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What's the algebraic trick to evaluate $\lim_{x\rightarrow \infty} \frac{x \sqrt{x}+\sqrt[3]{x+1}}{\sqrt{x^{3}-1}+x}$? $$\lim_{x \rightarrow \infty} \frac{x \sqrt{x}+\sqrt[3]{x+1}}{\sqrt{x^{3}-1}+x}$$
I got the first half:
$$\frac{x\sqrt{x}}{\sqrt{x^{3}-1}+x}=\frac{x\sqrt{x}}{\sqrt{x^{3}(1-\frac{1}{x^3})}+x}=\frac{1}{\... | Note that
$$\frac{x \sqrt{x}+\sqrt[3]{x+1}}{\sqrt{x^{3}-1}+x}=\frac{\sqrt{x^3}}{\sqrt{x^{3}}}\frac{1+\sqrt[6]{\frac{(x+1)^2}{x^9}}}{\sqrt{1-1/x^3}+1/\sqrt x}\to \frac{1+\sqrt{0}}{\sqrt{1-0}+0}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
On isomorphisms of the group of unit modulo n Let $U(n)$ denote the group of units of $\mathbb{Z}/n\mathbb{Z}$. I know that if $i$ and $j$ are relatively prime then $U(ij)$ is isomorphic to $U(i)\oplus U(j)$. I was wondering if the converse is true, namely if $U(ij)$ is isomorphic to $U(i)\oplus U(j)$ does this imply t... | The order of $U(n)$ is $\varphi(n)$ and we have $\varphi(ij)=\varphi(i)\varphi(j)$ if and only if $i$ and $j$ are coprime.
to see this notice that $\varphi(n)=n\prod\limits_{p|n} \frac{p-1}{p}$ so if a prime is repeated we have $\phi(ij)> \phi(i)\phi(j)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why doesn't this integral yield the area of the sphere? Here's how my book derives the formula for the volume of the sphere:
A sphere can be thought of as the solid of revolution generated by
revolving a semicircle about its diameter (see the figure below). If
the equation of the semi-circle is $x^2+y^2=a^2$, then... | Your set up
$$2\int_0^a{2\pi\sqrt{a^2-x^2}dx}$$
is not valid for the surface of the sphere since we should consider $\frac{dx}{\sin \theta}=\frac{dx}{ \frac{ \sqrt{a^2-x^2}} {a} }=\frac{a}{\sqrt{a^2-x^2}}dx$, then
$$S=2\int_0^a{2\pi a\,dx}=4\pi a^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Simplifying Quantified Statement For my assignment, I have to simplify this statement leaving no negations in the end.
$$\neg\exists x\ \forall x(\neg B(x) \wedge C(x))$$
Everything I've tried so far leaves me with a single negation sign on $B(x)$ or $C(x)$ and I just cannot figure this out.
| I assume that the negation on the very outside applies to the entire block.
What is the negation of a statement of the form $\exists x P(X)$? We should have $\forall x \neg P(x)$.
What is the negation of a statement of the form $\forall x Q(x)$? We should have $\exists x\neg Q(x)$.
Using these two rules, you can pass t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Homomorphisms of $\mathbb{F}_2$ that preserve $aba^{-1}b^{-1}$ Let $\mathbb{F}_2$ be the free group generated by $a$ and $b$. Suppose we are given a homomorphism $\phi: \mathbb{F}_2 \to \mathbb{F}_2$ with the property that $\phi(aba^{-1}b^{-1}) = aba^{-1}b^{-1}$. Can I conclude that $\phi$ is surjective? Can I concl... | It's a theorem of Nielsen known as "Nielsen commutator test" (Nielsen, J. Die Isomorphismen der aligemeinen unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78 (1918), 385–397.) stating that automorpisms of $F$ = $\langle x, y \rangle$ are precisely those endomorphisms which take $[x, y]$ to any conjugate or inverse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2709933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 0
} |
Proof that $-(-A) = A$ I have tried to prove that, given a set $A$, $-(-A)=A$. Are there flaws in my logic? I am very new to proof writing and set theory so any tips on structuring the proof would be greatly appreciated. My proof is:
If $S$ is the space and $A\subset S$, then $-(-A)=S-(-A)$.
Suppose that $x \in S-(-A)... | An easy way out:
$$x\in A\implies x\notin -A\implies x\in-(-A)\implies A\subset -(-A)$$
and on the other hand
$$x\in-(-A)\implies x\notin -A\implies x\in A\implies -(-A)\subset A$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Zero conditional mean and zero correlation $\newcommand{\Cov}{\mathrm{Cov}}$
$\newcommand{\E}{\mathrm{E}}$
Is $E(Y|X)=0$ equivalent to $\Cov(X,Y)=0$?
I know $E(Y|X)=0$ implies $\Cov(X,Y)=0$, because
$\Cov(X,Y) = \E(XY) - \E(X)\E(Y) = \E[\E(XY|X)]-E[X]E[E(Y|X)]=0$
But is the other way around true? Does $\Cov(X,Y)=0$ im... | No.
Let $Y=c$ a.s. with $c\neq0$.
Then $\mathsf{Cov}(X,Y)=0$ but $\mathsf E(Y\mid X)=c\neq0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A partition of 186 into five parts under divisibility constraints The sum of 5 positive natural numbers, not necessarily distinct, is 186. If placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and only if they have a common divisor greater than 1 (that is, they are not... | A Mathematica search finds and confirms:
$\{ 33, 77, 35, 15, 26 \}$
mylist = Select[
DeleteDuplicates /@
Select[IntegerPartitions[186, {5}], ContainsNone[#, {1}] &],
Length[#] == 5 &];
fulllist = Flatten[Permutations /@ mylist, 1];
Select[fulllist, (GCD[#[[1]], #[[2]]] > 1 &&
GCD[#[[2]], #[[3]]] > 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Angle matching in a quadrilateral with a single unknown vertex Three vertices of a quadrilateral are known (green)
with the fourth vertex (red) is unknown. The angles from the unknown vertex to the other three are also known (green a and b in the diagram). I need to work out the single position for the fourth vertex w... | I'll use different element naming conventions. The given points are $A$, $B$, $C$, with $|\overline{AC}| = a$, $|\overline{BC}| = b$, $\angle ACB = \gamma$. The desired point is $C^\prime$, such that $\angle AC^\prime C = \alpha$ and $\angle BC^\prime C = \beta$.
TL;DR:
$$C^\prime = p A + q B + r C \tag{$\star$}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Passwords: Two 50-characters vs one 100-characters In this Information Security question, we discuss whether or not a $100$ character secret randomly-generated username is equivalent to a $50$ character secret randomly-generated username plus a $50$ character secret randomly-generated password.
This answer [now deleted... | The answer that you are referring to is incorrect, the total number of combinations will be $62^{50}\cdot 62^{50}=62^{100}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 5,
"answer_id": 1
} |
Showing that a direct product is non-cyclic We have that $Z_{2} \times Z_{2}$ is non cyclic this can be easy seen by that $(2,2) \neq 1$ or simply by writing out the table, but I am searching for another method which I was introduced in during class. If I remember correctly it had something to do with LCM and perhaps L... |
If $\gcd(m,n)>1$ then $C_m \times C_n$ is not cyclic.
Indeed, let $L=\operatorname{lcm}(m,n)$. Then $g^L = 1$ for all $g \in C_m \times C_n$. Since
$$
L = \frac{mn}{\gcd(m,n)} < mn
$$
there is no element of order $mn$ in $C_m \times C_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
prove that $x(t) \in ]0,\pi[$ Given the Cauchy Problem:
$
\left\{
\begin{array}{@{}l}
x'(t) = \sin (x(t)),\ t\in\mathbb{R}\\
x(0)=x_0 \in ]0,\pi[
\end{array}
\right.
$
I try to prove that $x(t) \in ]0,\pi[ $
What I did:
Using Cauchy-Lipschitz theorem, I proved that the Cauchy Problem has a unique solution on $\mathbb{... | Below is the explicit solution. Probably that will help you investigate the values of $x(t)$.
$$x'(t)=\sin x(t)\Rightarrow x''(t)=\cos x(t)\cdot x'(t)=\cos x(t)\sin x(t)$$
On the other hand
$$x'(t)^2=\sin x(t) x'(t)\Rightarrow \int x'(t)^2\,dt=\int \sin x(t) x'(t)\,dt$$
But
$$\int \sin x(t) x'(t)\,dt=\int\sin x(t)\,dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Average life expectancy..exponential function Let
$$N_0 = \text{initial number of AIDS patients}$$
$$N= \text{number of patients left}$$
The equation is given by: $$N=N_0\exp(-kt)$$
What is the average life expectancy of one person?
(The answer is $t= \frac1k$)
How did we get to this answer without using expected value... | The answer is not very rigorous since, as you know, the DE itself is derived using expectation and average out everything but I think you will get the overall idea.
Let $N(\tau)=N_0-n$, $N(\tau+\delta)=N_0-(n+1)$.
Hence, one life has lapsed in time $\delta$.
$$\tau=\frac{1}{k}\ln\frac{N_0}{N_0-n}$$
$$\tau+\delta=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$x_{0}= \cos \frac{2\pi }{21}+ \cos \frac{8\pi }{21}+ \cos\frac{10\pi }{21}$ Prove that $x_{0}= \cos \frac{2\pi }{21}+ \cos \frac{8\pi }{21}+ \cos\frac{10\pi }{21}$ is a solution of the equation
$$4x^{3}+ 2x^{2}- 7x- 5= 0$$
My try:
If $x_{0}$, $x_{1}$, $x_{2}$ be the solutions of the equation then
$$\left\{\begin{matri... | HINT:
The polynomial $4x^{3}+ 2x^{2}- 7x- 5$ factors as $(x + 1) (4 x^2 - 2 x - 5)$.
So we need to prove that $x_{0}= \cos \frac{2\pi }{21}+ \cos \frac{8\pi }{21}+ \cos\frac{10\pi }{21}$ is a root of $(4 x^2 - 2 x - 5)$. What is the other root? It is $x_1=\cos \frac{4\pi }{21}+ \cos \frac{16\pi }{21}+ \cos\frac{20\pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Rudin's proof of Fatou's Lemma I have a question about the proof of Fatou's Lemma in Rudin's Real and Complex Analysis, 3rd ed, on page 23. I underlined the part that I don't follow in red below:
To prove the lemma, I would have to replace the underlined part by:
Then $g_k \le f_n, \forall n\ge k$, so that
$$\int_... | It is that $\lim_{k}g_{k}=\liminf_{n}f_{n}$ and we have $\displaystyle\int\lim_{k}g_{k}=\lim_{k}\int g_{k}$ by Monotone Convergence Theorem.
Now $\displaystyle\int g_{k}\leq\int f_{k}$ and taking limit infimum $k\rightarrow\infty$ both sides we have $\liminf_{k}\displaystyle\int g_{k}\leq\liminf_{k}\int f_{k}$. But $\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Negative of a power of a norm. I have one silly doubt if $\|.\|$ is a norm on a Hilbert space, then is it correct that $$\|x\|^{\mu} = \|-x\|^{\mu}, \qquad \mu \in (0, 1)$$
Please help me to understand the above concept.
According to me, it must be correct as $\|\alpha x \| = |\alpha|\|x\|$, for any scaler $\alpha$.
|
We have for a norm $\|.\|$ in a Hilbert-space and $\mu\in(0,1)$
\begin{align*}
\color{blue}{\|-x\|^{\mu}}=\|(-1)\cdot x\|^{\mu}=\left(|-1|\|x\|\right)^{\mu}=\color{blue}{\|x\|^{\mu}}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Volume of Ellipsoid using Triple Integrals Given the general equation of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$, I am supposed to use a 3D Jacobian to prove that the volume of the ellipsoid is $\frac{4}{3}\pi abc$
I decided to consider the first octant where $0\le x\le a, 0\le y \le b, 0... | HINT
Let use spherical coordinates with
*
*$x=ra\sin\phi\cos\theta$
*$y=rb\sin\phi\sin\theta$
*$z=rc\cos\phi$
and with the limits
*
*$0\le \theta \le \frac{\pi}2$
*$0\le r \le 1$
*$0\le \phi \le \frac{\pi}2$
Remember also that in this case
$$dx\,dy\,dz=r^2abc\sin \phi \,d\phi \,d\theta \,dr$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Differentiable function such that $\lim_\limits{x\to \infty}f(x)=\infty$
Let $f:(1, \infty) \to \mathbb{R}$ be a differentiable function such that $$f'(x)=\frac{x^2-(f(x))^2}{x^2(1+(f(x))^2)}$$
for all $x>1$. Prove that $\lim_\limits{x \to \infty}f(x)=\infty$
From the given relation, I got that $f$ is infinitely di... | Claim 1: $f$ is unbounded above.
Proof:
By contradiction. Suppose $f(x)\le M$ for all $x\in(1,\infty)$. Then $$
(1+f(x)^2)f'(x)=1-\frac{f(x)^2}{x^2}\ge\frac12,\quad x\ge M\sqrt2.
$$
Integrating
$$
f(x)+\frac13\,f(x)^3\ge \frac{x}{2}+C,\quad x\ge M\sqrt2.
$$
The function $h(u)=u+u^3/3$ is strictly increasing and has an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing inequality: $pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$? How can I show that $$pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$?
I've been stuck on this for a long time; I tried expanding out the taylor series on either side, and I... | Note
$$
\begin{align*}
p e^{x(1-p)}+(1-p)e^{-xp}
= e^{-xp}(1-p+p e^{x})
\le e^{(e^x-1-x)p}.
\end{align*}
$$
(Using the standard $1+x\le e^x$ for $x\in\mathbb R$.)
Now
$$
(e^x-1-x)
=\sum_{k\ge2}x^k/k!
\le x^2\sum_{k\ge2}1/k!=x^2(e-2)\le0.72 x^2\le \frac{3}{4}x^2.
$$
(where we used $0\le x\le 1$.)
Putting the two togethe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2711891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Fourier transform of Singular Function $f(x)$ with $\frac{1}{x^2}$ as $x\to 0^{+}$ and $\frac{1}{x^2} +\frac{1}{x}$ as $x\to 0^{-}$ In Lighthill's `An Introduction to Fourier Analysis and Generalised Functions', the Fourier transform of a function $f(x)$ is defined as:
$$
\mathscr{F}[f](y) \ = \ \int_{-\infty}^{\infty}... | First you need to define $\Theta(-x)/x$. It has to be defined as a functional, since it's not in the space of well-behaved test functions. The natural definition is
$$\left( \frac {\Theta(-x)} x, \phi \right) =
\int_{-\infty}^{-1} dx \frac {\phi(x)} x +
\int_{-1}^0 dx \frac {\phi(x) - \phi(0)} x,$$
which is the distri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Can $7n + 13$ ever equal a square? If not, why not? Can it be proved? And if it can be proved that it does equal a square (which I doubt), what is the smallest value for which this occurs?
| \begin{align}
1^2 &\equiv 1\pmod 7 \\
2^2 &\equiv 4\pmod 7 \\
3^2 &\equiv 2\pmod 7 \\
4^2 &\equiv 2\pmod 7 \\
5^2 &\equiv 4\pmod 7 \\
6^2 &\equiv 1\pmod 7 \\
7^2 &\equiv 0\pmod 7 \\
8^2 &\equiv 1\pmod 7 \\
9^2 &\equiv 4\pmod 7 \\
10^2 &\equiv 2\pmod 7 \\
11^2 &\equiv 2\pmod7\\
12^2 &\equiv 4\pmod7 \\
& \space\space\spa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
If two measures coincide on bounded continuous functions, do they coincide on Borel subsets? Let $X$ be a Hausdorff space, endowed with the Borel $\sigma$-algebra, and let $\mu, \nu$ be two regular Borel probabilities on $X$. Assume that $\int _X f \ \mathrm d \mu = \int _X f \ \mathrm d \nu$ for all $f \in C_b (X)$ (t... | It is true with perfectly normal spaces at least.
Let $\mathbb{B}(X)$ be all bounded measurable functions.
Take $\mathcal{H}=\{f\in \mathbb{B}(X)\ :\ \int fd\mu=\int fd\nu\}$.
$\mathcal{H}$ is a vectorial space which contains $1$ and is closed by monotone limits : if $(f_n)_{n\in\mathbb{N}}\subset\mathcal{H}$ and $f_n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If we select a random integer number of the set $[1000000]$ what is the probability of the number selected contains the digit $5$? If we select a random integer number of the set $[1000000]$ what is the probability of the number selected contains the digit $5$?
My work:
We know the sample space $S:$"The set of number o... | Notice that $P(\text{contains 5})=1-P(\text{doesn't contain 5})$. The latter is easily calculated by calculating the probabilities that each individual digit is not 5. Hence,
$$
\begin{align}
P(\text{doesn't contain 5})&=\left(\frac{9}{10}\right)^6\\
&=\frac{531441}{1000000}
\end{align}
$$
and
$$
\begin{align}
P(\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Different balls in an urn There are 20 red, 20 green and 20 blue balls in an urn.
*
*In how many different ways can 10 balls be selected?
*How many ways are there if there are 6 red balls instead of 20?
I would think to solve this with the binomial coefficient (i.e. 60 choose 10) but this seems far too simplistic ... | If you use stars and bars, the questions is how man ways can $10$ indistinguishable balls be place in $3$ buckets labelled "red," "green," and "blue" and the answer is $$\binom{10+3-1}{3-1} = \binom{12}{2}$$
For the second part of the question, we have the restriction that at most $6$ balls can be placed in the bucked... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $\frac{a}{a+ b}+ \frac{b}{b+ c}+ \frac{c}{2c+ 5a}\geq \frac{38}{39}$ $$a, b, c \in \left [ 1, 4 \right ] \text{ }
\frac{a}{a+ b}+ \frac{b}{b+ c}+ \frac{c}{2c+ 5a}\geq \frac{38}{39}$$
This is an [old problem of mine in AoPS]
(https://artofproblemsolving.com/community/u372289h1606772p10020940).
First solution
$$a... | You want to minimize the smooth function $F(a,b,c) = {\frac {a}{a+b}}+{\frac {b}{b+c}}+{\frac {c}{2\,c+5\,a}}$ over the cube $1 \le a,b,c \le 4$. The candidates for minimizer are critical points in the interior, points on a face where the gradient is orthogonal to the face, points on an edge where the gradient is orth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2712732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proof of Universal Mapping Property for tensor product of vector spaces Let $V$ and $W$ be two vector spaces. We define the tensor product of $V$ and $W$, denoted by $V \otimes W$, like wikipedia does that (https://en.wikipedia.org/wiki/Tensor_product). I want to prove the UMP. It is: if
$$
\begin{array}{llccl}
\pi & :... | For existence define $\tilde{l}(v\otimes w):=l(v,w)$ and you extend this linearly. This means you furthermore define $\tilde{l}(a\otimes b+c\otimes d):=\tilde{l}(a\otimes b)+\tilde{l}(c\otimes d)$ and $\tilde{l}(\lambda(v\otimes w)):=\lambda\tilde{l}(v\otimes w)$. Then check that this is well-defined and $l=\tilde{l}\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Approximation by biholomorphisms Assuming two domains $\Omega_1 \subset \Omega_2 \subset \mathbb{C}$ satisfy the criteria in Runge's theorem. So we know that any holomorphic function $f: \Omega_1\to \Omega_1$ can be approximated uniformly on compacts by (the restriction to $\Omega_1$ of) a sequence of holomorphic funct... | No. Counterexamples abound, obtained by choosing things so that $\Omega_2$ has a small automorphism group.
For example, let $\Omega_1$ be the right half plane, $\Omega_2=\Bbb C$, and $f(z)=1/z$. Recall that every biholomorphic map from $\Bbb C$ to itself has the form $z\mapsto az+b$; it's clear that such things cannot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is $\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$? How would you work something like this out?
Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?
| Hint: Given proper domain for the function so that $\cos(x) >0$ we can write:
$$(\cos x)^{\cos x} = e^{(\cos x) \ln(\cos x)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
How do i computed the Groebner Basis for this ideal? I have the ideal
$$\begin{split}I_{k} &=
\langle\,\, x_{1}^{3}-1,\\
&\qquad x_{1}^{2}+x_{1}x_{2}+x_{2}^{2},\\
&\qquad x_{1}^{2}+x_{1}x_{3}+x_{3}^{2},\\
&\qquad x_{2}^{3}-1,\\
&\qquad x_{2}^{2}+x_{2}x_{3}+x_{3}^{2},\\
&\qquad x_{3}^{3}-1 \rangle \end{split}$$ with le... | In general, what you should do is to take the $S$-polynomial for pairs and compute the remainder modulo all of the other polynomials.
For example, for the first two polynomials, $x_1^3-1$ and $x_1^2+x_1x_2+x_2^2$, compute the $S$ polynomial by multiplying to make the lcm of the leading coefficients:
$$
1(x_1^3-1)-x_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Stokes’ Theorem for arbitrary surface and boundary curve in $xz$-plane? I am tasked to find $\iint (\nabla \times {\bf V}) \cdot d{\bf S}$ for any surface whose bounding curve is in the $xz$-plane, where ${\bf V} = (xy + e^x) {\bf i}+ (x^2 -3y){\bf j} + (y^2 + z^2) {\bf k}$. I have attempted this via two methods and am... | You are correct. The answer is zero and you're almost there. Essentially you get zero because the differential form you're left with has a potential function.
$e^x dx + z^2 dz$ can be integrated and becomes $f(x,z) = e^x + z^3/3$.
Take an arbitrary parameterization, say $r (t) =(x (t), 0, z (t) )$, then plugging that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Subquotients of Module In Rotman's book on homological algebra (page 625), he says that if you have modules $Y\subset X\subset Z$ with $X/Y=Z$ then $Y=0$ and $X=Z$. It's not clear what he means by $X/Y=Z$ in the first place, but I can only assume he means isomorphic, although there then seems to be fairly easy counter-... | What Rotman wants to prove is that if $E^r= E^{r+1}$ in a spectral sequence, then $Z^{r+1} = Z^r$ and $B^{r+1} = B^r$. Remember that $E^{r+1}$ is computed as the homology of $E^r$ with respect to a differential. What Rotman is saying here is: the differential is zero iff $E^r=E^{r+1}$, iff the cycles (resp. boundaries... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
standard deviation of multiple vectors I have multiple vectors of length N, and I want to calculate the standard deviation (Std) of them, so if I have these vectors:
[1.03132, 1.456758,1.1324684]
[0.46546, 3.232658,3.1456444]
[0.21346, 0.568748,1.5554487]
The standard deviation vector is:
[0.34198709 1.10748961 0.8666... | The standard deviation measures how far from the average something is.
If you take the standard deviation of the entries of your result vector, you'll get a number that indicates how far each coordinate's standard deviation is from the average of your result vector's coordinates.
So for instance, if you result vector w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simon's Favorite Factoring Trick Problem Suppose that $x,y,z$ are positive integers satisfying $x \le y \le z$, and such that the product of all three numbers is twice their sum. What is the sum of all possible values of $z$?
Since this is for only positive integers, and there are sums and products involved, I think th... | Suppose $x,y,z$ are positive integers, with $x \le y \le z$, such that $xyz=2(x+y+z)$.
\begin{align*}
\text{Then}\;\;&xyz=2(x+y+z)\\[4pt]
\implies\;&xyz \le 2(3z)\\[4pt]
\implies\;&xy \le 6\\[4pt]
\implies\;&x^2 \le 6\\[4pt]
\implies\;&x \le 2\\[4pt]
\end{align*}
Consider two cases . . .
Case $(1)$:$\;x=1$.
\begin{ali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\frac{pq}{(p-1)(q-1)} < 2$ for distinct odd primes $p,q$ I need this lemma for another proof I'm doing, but I can't crack it. I want something of the structure:
$$\frac{pq}{(p-1)(q-1)} < \dots = \frac{pq}{\frac{1}{2}pq} = 2,$$
but I can't figure out what to do with the denominator.
| Assume $p>q$:
$$
2(p-1)(q-1)-pq=pq-2p-2q+2>pq-4p+2=p(q-4)+2
$$
which is $>0$ for $q>3$. For $q=3$,
$$
2(p-1)(q-1)-pq=4(p-1)-3p=p-4>0
$$
Therefore $2(p-1)(q-1)>pq$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2713937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Derivative of the function $\sin^{-1}\Big(\frac{2^{x+1}}{1+4^x}\Big)$
Find y' if $y=\sin^{-1}\Big(\frac{2^{x+1}}{1+4^x}\Big)$
In my reference $y'$ is given as $\frac{2^{x+1}\log2}{(1+4^x)}$. But is it a complete solution ?
Attempt 1
Let $2^x=\tan\alpha$
$$
\begin{align}
y=\sin^{-1}\Big(\frac{2\tan\alpha}{1+\tan^2\alp... | $y = arcsin(\frac{2^{x+1}}{1+4^x})\\\implies y = arcsin (\frac{2.2^x}{1+(2^x)^2})$
consider $y = arcsin(\frac{2x}{1+x^2})$
here let $x = tan(\theta) \implies \theta = arctan(x)$
$y= arcsin(\frac{2tan(\theta)}{1+ tan^2(\theta)}) = arcsin(sin(2\theta)) = 2\theta = 2arctan(x)$
so in your given question, $y = arcsin(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
All integer solutions of $x^4 + y^4 + z^4 - w^4= 1995$ This question is in the book 'The thrill and challenge of precollege mathematics'. I intend to attack this problem using Fermat's Little Theorem (FLT).
Notice that each term on LHS must be either of the form $5k$ or according to FLT $5k +1$ but the if $x^4$ is of ... | You can use a stronger result about fourth powers: $$x^4 \equiv 0,1 \mod 16$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the the coefficient of $\,x^r\,$ in $\,(1+x+x^2)^n$ I want to be able to explicitly write it as $a_r = \dots $
When using multinomial theorem, I'm getting stuck at 2 conditions, but I'm not able to simplify from there.
I wrote $(1+x+x^2)^n =\displaystyle \sum_{a,b,c}^{a+b+c=n}\frac{n!}{a!b!c!}(1)^a(x)^b(x^2)^c = \... | Here is an easier method to solve this without complex analysis:
$\left(1+x+x^{2}\right)^{n}=\left(\frac{1-x^{3}}{1-x}\right)^{n}=\left(1-x^{3}\right)^{n}(1-x)^{-n}$
Now,
$\begin{aligned}\left[x^{r}\right]\left(1+x+x^{2}\right)^{n} &=\left[x^{r}\right]\left(1-{ }^{n} C_{1} x^{3}+{ }^{n} C_{2} x^{6} -\ldots\right )(1-x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Is it always possible to choose $x \in (a, b)$ s.t. $\int_a^{x}f=\int_x^{b}f$ I am working out a homework problem about Riemann Integrals and the question is as follows:
Suppose that $f$ is integrable on $[a, b]$, then $\exists \ x \in [a,
b] s.t. \int_a^{x}f=\int_x^{b}f$. Is it always possible to choose $x$
to be... | Yes, provided that $\int_a^b f\ne 0$.
For example, if $f(x)=x$, and $[a,b]=[-1,1]$, then $\int_{-1}^x t\,dt=\frac{1}{2}(x^2-1)$, while $\int_x^1 t\,dt=\frac{1}{2}(1-x^2)$ and
$$
\int_{-1}^x t\,dt=\int_x^1 t\,dt\quad\Longrightarrow\quad x=\pm 1,
$$
and hence $x\not\in (-1,1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A sequence $p_n(x)$ that converges for infinitely many values of $x$
Let $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}, (c_n)_{n \geq 1}$ be sequences of real numbers. Knowing that the sequence $$p_n(x)=(x-a_n)(x-b_n)(x-c_n)$$
converges for infinitely many values of $x$, prove that it converges for every $x \in \mathbb{R}$.
... | Put $p_n(x)=x^3+A_nx^2+B_nx+C_n$.
Look at $\frac{p_n(r)-p_n(s)}{r-s}=r^2+rs+s^2+A_n(r+s)+B_n$. This one should also converge for infinitely many $r,s$.
Therefore,$A_nx+B_n$ converges for infinitely many $x$. Taking differences again we get that $A_n$ converges. Therefore $B_n$ converges, and $C_n$ converges.
If follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Noetherian module and noetherian ring If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?
| Let $P_n$ be the $R$-sub-module of $R[x]$ whose elements are all polynomials of degree at most $n$.
Then you have the infinite chain
$$P_0 \subset P_1 \subset P_2 \subset \cdots$$
with all inclusions proper.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Solve recurrence $T(n) = T(\frac34n) + \sqrt n$ How can I solve the following recurrence relation?
$T(n) = T(\frac34n) + \sqrt n$
I have attempted to solve it with the substitution method.
I believe that the general pattern is $T(n) = T((\frac34)^kn) + n^{\frac1{2^k}}$,
but I cannot think of what method I need to follo... | An elementary way which doesn't use the Master Theorem could be:
$\begin{align*}
T(n) = T\left(\frac{3}{4} n\right) + \sqrt{n} \iff T(4n^2) = T(3n^2) + 2n
\end{align*}$
Thus, let us assume that $T(n) = C n^{\alpha}$ for some $C, \alpha$, is solution for the previous relation.
Then: $C(4^{\alpha} - 3^{\alpha})n^{2\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Application Chinese Remainder Theorem to Dedekind Domains I have a question about the application of CRT in a proof of following thread:
Dedekind domain with a finite number of prime ideals is principal
The claim is that a Dedekind domain with a finite number of prime ideals is already principal:
In his answer @pki use... | First $\newcommand{\pp}{\mathfrak{p}}\newcommand{\mm}{\mathfrak{m}}(x)=\pp_1^{e_1}\cdots \pp_n^{e_n}$ for some $e_i\in\newcommand{\NN}{\mathbb{N}}\NN$. If $e_i \ge 1$ for $i\ge 2$, then $x\in(x)\subset \pp_i$, however, $x\equiv 1 \pmod{\pp_i}$, so this is impossible. Hence $(x)=\pp_1^{e_1}$, for some $e_1\ge 1$ (since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Every subspace of a Frechet-Urysohn space is a Frechet- Urysohn space and hence also a sequential space I'm trying to understand this statement:
Every subspace of a Frechet-Urysohn space is a Frechet- Urysohn space and hence also a sequential space.
Definition of Frechet-Urysohn space:
A topological space $(X, \tau)... | I suspect you are somehow misunderstanding the statement or the definition of Frechet-Urysohn, since the case of a singleton is really quite trivial. Hopefully the following explanation will help you find your misunderstanding.
If $X=\{x\}$ is a singleton, there are only two subsets of $X$: either $S=\emptyset$ or $S=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
discussing the series i want to prove the divergence of the infinite series $\sum_{n=0}^\infty \frac{(-1)^n x^n}{(n+1)^p}$ it's an alternating series so we will be dealing with the series $\sum_{n=0}^\infty \frac{A_n x^n}{(n+1)^p}$ i tries using leibnitz test but can only prove weather it converges or not
so is it po... | Note that
$$\sum_{n=0}^\infty \frac{(-1)^n x^n}{(n+1)^p}$$
by ratio test
$$\left|\frac{(-1)^{n+1} x^{n+1}}{(n+2)^p}\frac{(n+1)^p}{(-1)^n x^n}\right|=|x|
\left(\frac{n+1}{n+2}\right)^p\to |x|$$
thus the series
*
*converges for $|x|<1$
*converges for $x=1$ by Leibniz
*for $x=-1$ by limit comparison test converges f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding functional equation in which $g(1)=0$ and $g'(1)=1$
Let a function $g:\mathbb{R^{+}}\rightarrow\mathbb{R}$ is a differentiable function such that $2g(x)=g(xy)+g\bigg(\frac{x}{y}\bigg)\forall x,y\in\mathbb{R^+}$ and $g(1)=0$ and $g'(1)=1$. Then $g(x)$ is
Try: Differentiate both side with respect to $x$, treat... | Something more general may be proved.
A continuous function $g:\mathbb{R^{+}}\rightarrow\mathbb{R}$ with $g(1)=0$ satisfies $2g(x)=g(xy)+g\bigg(\frac{x}{y}\bigg)\forall x,y\in\mathbb{R^+}$ iff $g= c \ln$ for some real constant $c$.
Obviously $g=c\ln$ satisfies the functional equation and $g(1)=0$.
On the other hand th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the sum of all of the odd divisors of $6300$? What is the sum of all of the odd divisors of $6300$?
Hello! I am a middle school student, so simply-worded answers would be super helpful.
To solve this, I tried to find the sum of the odd divisors of a few smaller numbers, like 10. I know $10 = 2 * 5$, so I thoug... | $$6300 = 2^2\cdot3^2\cdot 5^2\cdot 7$$
The set of odd factors of $6300$ is equal to the set of factors of $3^2 \cdot 5^2 \cdot 7$.
The sum of factors of $3^2 \cdot 5^2 \cdot 7$ is
\begin{align}
\sum_{a=0}^2 \sum_{b=0}^2 \sum_{c=0}^1 3^a\cdot 5^b \cdot 7^c &= \sum_{a=0}^2 3^a\sum_{b=0}^2 5^b\sum_{c=0}^1 7^c \\
&=(1+3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How was the area formula for a circle ($A = \pi r^2$) derived before the introduction of calculus? How did mathematicians prior to the coming of calculus derive the area of the circle from scratch, without the use of calculus?
The area, $A$, of a circle is $\pi r^2$. Given radius $r$, diameter $d$ and circumference $c$... | By trial and error and by numerical approximations. The ancient mathematicians (Babylonians, 1800 BC) tried to square a circle (approximating the area of circle with a square, constructing a square with the same area as a circle, proved impossible in 1882 AD), to calculate $\sqrt{2}$, $\pi$, etc.
The precise calculatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 11,
"answer_id": 6
} |
Proving limits for fractions using epsilon-delta definition Using the $\epsilon - \delta $ definition of the limit, prove that: $$\lim_{x\to 0} \frac{(2x+1)(x-2)}{3x+1} = -2$$
I firstly notice that my delta can never be greater than $\frac{1}{3}$ because there is a discontinuity at $x=-\frac{1}{3}$.
I applied the stand... | Let $x > -\dfrac13$ and $|x| < \delta$, then $2x+3, 3x+1 > 0$, where $\delta > 0$ is to be determined.
$$\frac{2x+3}{3x+1} \le \frac{2\delta+3}{\underbrace{1-3\delta}_{\mbox{take $\delta < \frac13$}}}$$
Take $\delta < \dfrac13$ so that the denominator is positive. Observe that when $|x| < \delta$, the fraction is posi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2715873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Dual Of Integer Network Formulation I have the following IP and I wonder how to write the dual of it
as a network flow problem:
\begin{align}
\max & \sum_{i \in N} w_ix_i \\[4pt]
\text{s.t. } & x_i \leq x_j, \forall (i,j) \in A \\[10pt]
& 0 \leq x_i \leq 1, \forall i \in N
\end{align}
I was thinking that the dual could... | There are three errors: (1) Because $y_{ij}$ is associated to the constraint where the right hand side is 0 ($x_i - x_i \leq 0$), it should not appear in the objective. (2) The coefficients $w_i$ seem to be missing in the dual. (3) You currently do not have a dual variable associated to the constraint $x_i \leq 1$.
Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim_{x\to \infty } (x +\sqrt[3]{1-{x^3}} ) $ $$\lim_{x\to \infty } (x +\sqrt[3]{1-{x^3}} ) $$
What method should I use to evaluate it. I can't use the ${a^3}$-${b^3}$ formula because it is positive. I also tried to separate limits and tried multiplying with $\frac {\sqrt[3]{(1-x^3)^2}}{\sqrt[3]{(1-x^3... | I usually suggest to make the substitution $x=1/t$, so the limit becomes
$$
\lim_{t\to0^+}\frac{1+\sqrt[3]{t^3-1}}{t}=
\lim_{t\to0^+}\frac{1-\sqrt[3]{1-t^3}}{t}=
\lim_{t\to0^+}\frac{1-1+\frac{1}{3}t^3+o(t^3)}{t}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 2
} |
Integral $\int \frac{\cos x }{2+\sin 2x} dx$ I have tried to find antiderivative of
$$ \frac{\cos x\ }{2+\sin 2x} $$
using the variable change $t= \cos x -\sin x$ with sin $\sin2x=2\sin x\cos x $. But i don't come up to its closed-form result as shown below.
How can I find its antiderivative? Thanks in advance
| You can do the change $t=\dfrac{1+\sin(x)}{\cos(x)}$ to arrive to $\displaystyle \int\dfrac{2t\mathop{dt}}{t^4+2t^3+2t^2-2t+1}$
I'm joking, in fact this comes from successive changes:
*
*$\displaystyle u = \sin(x)\quad\to\quad\int\dfrac{\mathop{du}}{2+2u\sqrt{1-u^2}}$
*$\displaystyle \tanh(v)=u\quad\to\quad\int\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Could a functional be defined to be with compact support? Could a functional, $F:C^\infty\to\mathbb R$ be defined to have the properties of rapidly decreasing and/or with compact support, just like the real-valued functions?
| No. In order to understand why, imagine that $F$ had compact support $K \subset C^\infty (X)$. Let $f \in K$. It follows that $F(f) \ne 0$, so by the continuity of $F$ there must exist a whole neighbourhood $U$ of $f$ in $C^\infty (X)$ on which $F \ne 0$, which means that $U \subseteq K$. This means that $f$ has $\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem evaluating a contour integral using parametrization I tried to solve the following contour integral:
$$
\oint_\gamma {\frac{{dz}}{{z - c}}}
$$
Where $\gamma$ is a disk centered at the origin. In order to do so, I used the following parametrization:
$$
\begin{array}{l}
z &= Re^{i\varphi }, \qquad 0 < \left| R... | You are assuming that there is a differentiable function $\ln$ from $\mathbb{C}\setminus \{0\}$ into $\mathbb C$ such that $\ln'(z)=\frac1z$. There isn't.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Proving Inequalities With Mathematical Induction I'm currently working on this problem:
$$ 1 + 2^n + ≤ 3^n \text{ for all } ≥ 1 $$
So far, I have:
Basis Step:
$ 1 + 2^1 ≤3^1 $
$ P(1) \text{ is true} $
Inductive Step: Assume P(k) holds, prove P(k+1)
$P(k) = 1 + 2^k ≤ 3k$
$ P(k+1) = 1 + 2^{k+1} ≤ 3^{k+1} \text{ (I.H.)}... | \begin{align*}
1+2^{k+1}&=1+2\cdot 2^{k}\\
&\leq 1+2\cdot(3^{k}-1)\\
&=2\cdot 3^{k}-1\\
&\leq 2\cdot 3^{k}\\
&\leq 3\cdot 3^{k}\\
&=3^{k+1}.
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2716960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
} |
Characterizing integral domains for which every ideal, that can be generated by two elements, is projective? Let $R$ be an integral domain. $R$ is called a Prufer domain if every finitely generated ideal of $R$ is projective. There are various equivalent conditions for $R$ being a prufer domain , in terms of ideal arit... | First, one sees immediately that if $R$ has the property that all two generated ideals are projective, the same holds for any localizations.
Then one shows by induction as follows, any $n$-generated ideal is projective assuming the result for smaller $n$. Clearly we can assume $n>2$ and let $I=(x_1,\ldots, x_n)$ and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proving that $\frac{f(b)-f(a)}{b-a}+ \left(\frac{g(b)-g(a)}{b-a}\right)^2\le \max_{t\in [a,b]}\{f'(t)+(g'(t))^2\}$ Let $f,g\in C^1([a,b])$ with $a<b$ then prove that
$$\frac{f(b)-f(a)}{b-a}+ \left(\frac{g(b)-g(a)}{b-a}\right)^2\le \max_{t\in [a,b]}\{f'(t)+(g'(t))^2\}$$
It smells like there is some mean value theore... | \begin{align*}
&\dfrac{f(b)-f(a)}{b-a}+\left(\dfrac{g(b)-g(a)}{b-a}\right)^{2}\\
&=\int_{a}^{b}f'(t)\dfrac{dt}{b-a}+\left(\int_{a}^{b}g'(t)\dfrac{dt}{b-a}\right)^{2}\\
&\leq\int_{a}^{b}f'(t)\dfrac{dt}{b-a}+\int_{a}^{b}(g'(t))^{2}\dfrac{dt}{b-a}\\
&\leq\max_{t\in[a,b]}\{f'(t)+(g'(t))^{2}\}\int_{a}^{b}\dfrac{dt}{b-a}\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Equilateral triangle inscribed in an ellipse An equilateral triangle of side length $ L\approx 6.14$ and one side inclination $ \approx 49.52^{\circ}$ is inscribed in an ellipse of semi-axes $(a,b) = (5,3)$. Drawn in Geogebra by Java mousing .. trial/error.
Are $ (L,\alpha) = f(a,b) $ in this configuration unique? I... | In following discussion, we will assume $a > b > 0$.
Identify the plane with complex plane through the map $\mathbb{R}^2 \ni (x,y) \mapsto z = x + iy\in \mathbb{C}$.
In terms of $z$, the equation of ellipse $\mathcal{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ becomes
$$A(z^2+\bar{z}^2) + Bz\bar{z} = 1\quad\text{where}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Proof by induction with factorials I need help with proving this:
$$\sum_{i=1}^n \frac{i-1}{i!}=\frac{n!-1}{n!}$$
My induction hypothesis is:
$$\sum_{i=1}^{n+1} = \sum_{i=1}^n \frac{i-1}{i!}+\frac{(n+1)!-1}{(n+1)!}=\frac{(n+1)!-1}{(n+1)}$$
I tried a few things and landed here:
$$\frac{(n+1)n!-1+n}{(n+1)n!}=\frac{(n+1... | "My induction hypothethesis is
$\sum_{i=1}^{n+1} = \sum_{i=1}^n \frac{i-1}{i!}+\frac{(n+1)!-1}{(n+1)!}=\frac{(n+1)!-1}{(n+1)}$"
WHY?!?!?!?!?!?!?!?
$\sum_{i=1}^{n+1}\frac {i -1}{i} = \sum_{i=1}^n \frac {i-1}{i} + \frac {n+ 1 - 1}{(n+1)!}$ and $\frac {n+1 - 1}{(n+1)!} \ne \frac{(n+1)!-1}{(n+1)!}$ And setting $n\to n+1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Double implication in natural language I'm talking about double implication like:
(P → Q) → Q
I know that this is equivalent to (P ∨ Q), but I don't quite understand why. Let's say I take proposition P to be "having guns", and proposition Q to be "violence", then I would express it in natural language as:
"If having gu... | The trouble is that the material implication $\rightarrow$ does not always perfectly match the English 'if ... then...'.
This mismatch is called the Paradox of Material Implication.
So, while given the mathematical definitions of the truth-functional operators $\rightarrow$ and $\lor$ it is true that $(P \rightarrow Q)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is $W(S_1,S_2)$ convex? Let $F=\mathbb{C}^4$ be endowed with the norm $\|\cdot\|_2$. Let the operators $$
S_1=\left(\begin{array}{cccc}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right)\;\mbox{and}\;\;S_2=\left(\begin{array}{cccc}0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)
.$$
The numerical range of $(S_1,S... | The open line segment is $\{(t/2,1-t)|0<t<1\}.$ Given $0<t<1,$ choose any $c,d$ such that $|c|^2+|d|^2=1-t$ for example, $c=0, d=\sqrt{1-t}$ and let $a=b=\sqrt{t/2}.$ Then $\overline{a}b=t/2, |a|^2+|b|^2+|c|^2+|d|^2=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove the center of the unit circle has the highest average ray length? If the average ray length is the average distance of the segments from a point inside the circle to points evenly distributed on the boundary:
Prove the center of the unit circle has the highest average ray length.
My Attempt
Convert the circle $x^... | By symmetry, the average distance from the boundary of $\|z\|\leq 1$ is a function of the distance from the origin. If $x\in(0,1)$ its average distance from the the boundary of the unit circle centered at the origin is given by
$$ \frac{1}{2\pi}\int_{0}^{2\pi}\sqrt{(x-\cos\theta)^2+\sin^2\theta}\,d\theta =\frac{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
What it means that a function depends on $x,y,u,u_x,u_y$? Definition: A PDE equation is quasilinear if $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu+G+\Phi(x,y,u,u_x,u_y)=0,$ where $A,B,C,D,E,F,G$ are functions of $x,y,u.$
Is this equation quasilinear? $$(x^2+u^2)u_x-xyu_y=u^3x+y^2$$
The answer is yes it is because $D=(x^2+u^... | Using the definition in the note:
A PDE is said to be quasilinear if it is linear in its highest derivative. That is, the
coefficient of highest order does not depend on any highest order partial derivative.
So the PDE $xu_x + yu_y +u^2 = 0$ IS quasilinear. Indeed in the note it is not claimed that the above is NOT... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2717901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Combinatorics: Find a recursive formula for partitions of integers into three partitions. I need to find a recursive formula for $p_n$ the number of ways to partition $n$ into three partitions. For example if we look for the partitions of $6$ then they are $1+1+4$, $1+2+3$, and $2+2+2$. Intuitively I look for the num... | You would better think of the problem as a star&bar setting, then you just need to divide by number of permutations that varie between $3!$,$3!/2$,$1$, but since this is more of complicated and overkill for this task, and not requirements-fit, I could sentence the recursive function of all available partitions $A+B+C$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Cross product in $\mathbb R^n$ (from Spivak's book) Spivak defines cross product in this way:
$\quad$ We conclude this section with a construction which we will restrict to $\mathbf{R}^n$. If $v_1,\ldots,v_{n-1}\in\mathbf{R}^n$ and $\varphi$ is defined by $$\varphi(w)=\det\pmatrix{v_1 \\ \vdots \\ v_{n-1} \\ w},$$ the... | Just for kicks, here's a proof of the Riesz representation theorem that Ivo talks about in the (much easier) finite dimension case.
Suppose $V$ is a real, finite dimensional inner product space. Let $f:V\rightarrow \mathbb{R}$ be a linear functional. We claim that there is a unique $z\in V$ for which $f(w) = \langle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
If $3x^2-6x+p=0$ has roots $\alpha$ and $\beta$, then find a quadratic with roots $(\alpha+\beta)/\alpha$ and $(\alpha+\beta)/\beta$ This one comes from an IGCSE Pure Math past paper:
The equation $3x^2-6x+p=0$ has roots $\alpha$ and $\beta$. Without solving the equation, form a quadratic equation with roots $\frac{\a... | Yes it is a correct way indeed
$$\left(x-\frac{\alpha+\beta}{\alpha}\right)\left(x-\frac{\alpha+\beta}{\beta}\right)=x^2+\left(\frac{(\alpha+\beta)^2}{\alpha\beta}\right)x+\frac{(\alpha+\beta)^2}{\alpha\beta}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Integration of $\sqrt{1-x^2}$ by parts. Interpretation of 2nd term. I was wondering about the anti-derivative of $\sqrt{1-x^2}$
Method for solving it is given here. This solution is fine, but I was bit confused about last term in first line i.e.
$$\int \left(\frac{d}{dx}\sqrt{1-x^2}\int dx\right)dx$$
$$=\int x\cdot\fra... | \begin{align}
I & = \displaystyle\int \sqrt{1-x^2} \\
& = x\sqrt{1-x^2}+\displaystyle \int \dfrac{x^2}{\sqrt{1-x^2}} \\
& = x\sqrt{1-x^2}+\displaystyle \int \dfrac{x^2+1-1}{\sqrt{1-x^2}} \\
& = x\sqrt{1-x^2}+\arcsin\left(x\right)-I \\
& =\dfrac{x\sqrt{1-x^2}}{2}+\dfrac{\arcsin\left(x\right)}{2}+C
\end{align}
EDIT... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Construction of the Legendre Polynomials by Gram Schmidt I'm stuck with a homework problem from my functional analysis class. The question is:
"Show that the Gram-Schmidt orthonomalization procedure in $L^2(-1,1)$, starting from the series $(x^n)^{\infty}_{n=0}$ provides an orthonomal basis, given by $(b^n)^{\infty}_{... | For any inner product there is indeed a unique (up to scaling) polynomial $f\in P_{n+1}(x)$ orthogonal to $P_n(x)$ (the space of polynomials of degree less than or equal to n).
To see this assume (seeking contradiction) that $f, g$ are linearly independent polynomials in the orthogonal compliment of $P_n(x) \leq P_{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find bases I am currently looking at the question:
Let $$A =\begin{bmatrix}
1& −1& 3& 1& 2\\
4& −4& 12& 6& 0\\
−3& 3& −9& −4& −2\end{bmatrix}$$
By bringing the matrix A into row echelon form, find bases for row(A), col(A) and N(A). Determine the rank and nullity of A, an... | We don’t need to proceed further, note also that
*
*a basis for $col(A)$ is given be the first and fourth vectors of the original matrix (corresponding to pivot columns in RREF)
*a basis for $row(A)$ is given by the two rows in the RREF
*to find the null space solve the system $Ax=0$ using A in RREF; since it has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Complex solutions of the equation $x^{\frac{1}{2}}+1=0$ How can I find complex solutions to the equation $$x^{\frac{1}{2}}+1=0$$
Squaring gives x=1 but it's not the solution
|
Squaring gives $\,x=1\,$ but it's not the solution
So you proved that $\,x^{1/2} = -1 \implies\, \big(x^{1/2}\big)^2 = (-1)^2 \implies x=1\,$, but $\,x=1\,$ does not verify the original equation (assuming that $\,x^{1/2}\,$ means the principal value of the complex square root). Therefore the original equation has no ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2718955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Prove that $x\mapsto e^x$ is continuous at $x_0 = 1$ ($\delta-\varepsilon$ proof) Prove that the function $$f(x)=e^x:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$ is continuous at $x_0=1$ using the delta epsilon definition of continuous, which is:
$$\forall \varepsilon >0 \exists \delta>0 (\forall x\in D:|x-x_0|<\de... | Here is some hints that will help you. It's just write it with more details.
Let $x_0\in\mathbb R$ and $\epsilon>0$ be arbitrary real numbers.
Notice that
$$|e^x - e^{x_0}| = |e^x - (1+x/n)^n + (1+x/n)^n - (1+{x_0}/n)^n + (1+{x_0}/n)^n - e^{x_0}| \leq |e^x-(1+x/n)^n| + |(1+x/n)^n-(1+{x_0}/n)^n| + |(1+{x_0}/n)^n-e^{x_0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Prove that $f^\ast\omega=\det f \cdot \omega$ Let $V$ be a vector space of dimension $n$ and $f: V\to V$ a linear operator. I need to show that $f^\ast:\Lambda^n(V)\to \Lambda^n(V)$ is multiplication by $\det f$.
My try:
Since $\dim \Lambda^n(V)$ is $1-$dimensional and $f^\ast$ is linear, $f^\ast$ must be multiplicati... | The second equality should be
$$\omega(f(v_1),\dots,f(v_n))=\det A \cdot \omega (v_1,\dots,v_n).$$
Let $w_i = f(v_i)$, we have
$$
w_i = f(v_i) = f(\sum_jv_{ji} e_j) = \sum_jv_{ji}f(e_j) = \sum_k\big(\sum_j A_{kj} v_{ji}\big)e_k = \sum_kw_{ki}e_k,
$$
Therefore by applying the theorem twice,
\begin{align}
\omega(w_1,\do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Two circles with two common outer tangents have same chords Let $\omega_1$ and $\omega_2$ be two circles, $r_1 < r_2$, they have two common outer tangents, let $A$ and $B$ - common points of first outer tangent and $\omega_1$ and $\omega_2$ respectively, $C$ and $D$ - common points of second outer tangent, $E$ and $F$ ... |
Join centers $G$ and $H$. Since $ABHG$ and $CDHG$ are congruent trapezoids, then tangents $AB$ and $CD$ are equal.
But $AB^2=BE\cdot BC$, and $CD^2=CF\cdot CB$ [Euclid III, 36].
Therefore,$$BE\cdot BC=CF\cdot CB$$and$$\frac{BE}{CB}=\frac{CF}{BC}$$making$$BE=CF$$Therefore$$CE=BF$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
set theory notation $\in \uparrow$ in set theory I'm reading the following set of notes
http://ozark.hendrix.edu/~yorgey/settheory/index.html
and on page 6 of the full set of notes (or first page of the second link), the symbol $\in \uparrow$ is used, though $\uparrow$ isn't quite right because its more like half that.... | The symbol in question is $\upharpoonright$, which is used for the restriction of a function or a relation to a subset of its domain. In particular:
*
*If $f : A \to B$ is a function and $U \subseteq A$, then $f \upharpoonright U : U \to B$ is the function defined by $(f \upharpoonright U)(a)=f(a)$ for all $a \in U$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why do people study algebraic extension? Yesterday, I learned Kronecker’s theorem and a finite extension. And now I’m studying the next chapter, Algebraic extension.
I think the next theorem shows how important algebraic extension is
Let $E$ be an extension field of a field $F$. Let $α$ be an element of $E$. If $α$ ... | I think the original motivation for studying field extensions was, as in the theorem you stated, to solve polynomials. One of the big results after a few lectures of algebraic field extensions is that every field can be embedded into a unique algebraically closed field, called the algebraic closure.
Actually, solving ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
PDF of sum of random variables (with uniform distribution) How can I solve this:
Random variables $X,Y$ ~ Unif$(0, 1)$ are independent. Calculate the probability density function of sum $X + 3Y$.
I couldn't find a sum for uniformally distributed random variables. I assume I have to go straight to the PDF and solve it ... | Easy Understanding of Convolution The best way to understand convolution is given in the article in the link,using that
I am going to solve the above problem and hence you could follow the same for any similar problem such as this with not too much confusion.
$Z = X+ 3Y$ where X and Y are U(0,1).
I am going to define... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
DGA formality via $A_\infty$ Given an $A_\infty$-quasi-isomorphism $A\rightarrow B$ of dgas A and B how does one get a zig-zag of dga-quasi-isomorphisms $A\rightarrow \cdot \leftarrow \ldots \leftarrow \cdot \rightarrow B$? This is one implication in an equivalence on page 7 in B. Vallette, ALGEBRA + HOMOTOPY = OPERAD ... | This essentially follows from the "rectification" procedure that is mentioned in the paper of Vallette (see e.g. Chapter 11 of the book Algebraic Operads of Loday and Vallette for more detail).
Given an augmented associative algebra $A$, you have the bar construction $BA$. It is the cofree conilpotent coalgebra on the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2719961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Interpret this notation (ODE) I need help how to interpret the notation in the following IVP:
\begin{align}
\dot x&=f(t,x), \quad \tag 1\\
x(t_o)&=x_0, \quad \tag 2
\end{align}
We assume $f\in C(U,\mathbb R^n)$, where $U$ is an open subset of $\mathbb R^{n+1}$ and $(t_0,x_0)\in U$.
Q1: Does it mean I actually have... | Your interpretation is the way most people I know would interpret that, and I can't see how else you could interpret it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
why is the solution to $x^2 = 3$ the same as $x = \pm \sqrt 3$ I understand that in order to simplify $x$ in this equation:
$x^2 = 3$ we would need to get the square root on both sides what i dont understand is the fact that it is written as $x = \pm \sqrt 3$.
I don't understand where the $±$ comes from , this proble... | It comes from the identity $\sqrt{x^2}=|x|$. Now, solve $|x|=\sqrt{3}$. Also, $x^2=3$; $x^2-3=0$; $(x-\sqrt{3})(x+\sqrt{3})=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Inequality involving cyclic sums (Muirhead? Schur? Something else?) Let's define:
$$M[a, b, c]=\sum_{cyc}x^a y^b z^c $$
I need to prove that for all positive and real $x$, $y$ and $z$:
$$M[6, 3, 0] + M[3, 3, 3] \ge M[5, 2, 2] + M[4, 4, 1]$$
From Muirhead's inequality it's obvious that $M[6,3,0]$ is the biggest. But $M[... | Let $\frac{a}{c}=x$, $\frac{b}{a}=y$ and $\frac{c}{b}=z$.
Hence, $xyz=1$ and after dividing of the both sides by $a^3b^3c^3$ we need to prove that
$$\sum_{cyc}\left(x^3-\frac{x}{y}-\frac{y}{x}+1\right)\geq0$$ or
$$\sum_{cyc}(x^3-x^2y-x^2z+xyz)\geq0,$$ which is Schur.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determinant of a matrix and linear independence (explanation needed) It is written on Wikipedia that:
$n$ vectors in $\mathbb R^n$ are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero
Can someone explain this to me? You do not have to give a compl... | The determinant is an n-linear (multilinear) alternating form.
(Let us assume that our determinants are with respect to an arbitrary base of the space considered, it's just a technical aspect, for rigor, but you should consider the determinant of a family with respect to a certain base.)
Alternating characteristic
What... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 9,
"answer_id": 3
} |
Stuck on Khan Academy Math Problem: Structure in Expressions I am having trouble understanding a math problem on Khan Academy even with the explanation they give me.
The expression:
$$({x^2} + {h^2})({x^2} - {h^2})
$$
can be written as
$$(1 + m - p){x^4} - mp
$$
where h, m, and p are constants what is one possible val... | I think it must be $$1+m-p=1$$ and $$mp=h^4$$ solving this we get $$m=\pm h^2$$ and $$p=\pm h^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question concerning exact sequences of $R$-modules Let $R$ be a commutative ring. I would like to prove the following two assertions :
(1) If $0 \longrightarrow X \overset{\alpha}{\longrightarrow} Z \overset{\beta}{\longrightarrow} Y$ is an exact sequence of $R$-modules, then $ker(\beta) \simeq X$.
(2) If $X \overset{... | For (1):
The exact sequence
$0 \longrightarrow X \overset{\alpha}{\longrightarrow} Z \overset{\beta}{\longrightarrow} Y \tag 1$
implies that
$\ker \alpha = \text{Im}(0) = 0, \tag 2$
thus $\alpha:X \to Z$ is injective, meaning
$\alpha:X \to \text{Im} \; \alpha \subset Z \tag 3$
is an isomorphism of $X$ to $\alpha(X) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Describing the Kernel of a map from The Fundamental Group to the integers If my group is the fundamental group with genus $g$ of surfaces
$S_g=⟨a_1,b_1,…,a_g,b_g\mid[a_1,b_1]...[a_g,b_g]=1⟩ $
and I have a map $H$ from my group to the integers:
$H: S_g\rightarrow\mathbb{Z}$
such that $a_1$ goes to $1$ and all other elem... | I guess, you mean all other generators, i.e. $b_1,a_2,\dots,b_q$, by 'all other elements' and 'everything except $a_1$'.
In your equation with $1-0+1$, the element $a_1$ can be replaced by any element of the group: $H(xb_1x^{-1})=0$. That is - such as every kernel - $\,\ker H$ will be a normal subgroup.
Specifically, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2720895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Expectation of stationary and ergodic process Suppose $\{y_t\}$ is stationary ergodic process and $\mathbb{E}[y_t|y_{t-j}, y_{t-j-1}, ...] \rightarrow _{m.s.} 0$ as $j \rightarrow \infty$.
Is $\mathbb{E}[y_t] = 0?$
I tried to find a counterexample for this statement, but failed. Thus it seems that this is correct, but... | Stationarity implies that $Ey_t$ is independent of $t$. Since $E(y_t|t_{t-j},...) \to 0$ in the mean we get $Ey_t \to 0$. Hence $Ey_t=0$ for all $t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Dual basis with non-degenerate bilinear form Let $V$ be $n$-dimensional vector space and $\{x_1,\cdots,x_n\}$ a basis. Let $\beta:V\times V\rightarrow F$ be a non-degenerate (symmetric) bilinear form. This implies that there exists a dual basis $\{y_1,\cdots, y_n\}$ of $V$ w.r.t. $\beta$, i.e.
$$\beta(x_i,y_j)=\delta_... | Let $B$ be the matrix with entries $b_{i,j}=\beta(x_i,x_j)$. It's symmetric
and invertible. Write $C=B^{-1}=(c_{i,j})$. Then $y_j=\sum_kc_{j,k}x_k$.
One checks
$$\beta(x_i,y_j)=\sum_k c_{j,k}b_{i,k}=\delta_{i,j}$$
using the fact that $CB^t=CB=I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding the fixed point.. I don't quite understand this question.
$f(x)=ax^2+bx+c$, where $a=14,b=−25$ and $c=−7$. There are two fixed points at $x_1$
and $x_2$, where $x_1>x_2$. How to get the value of $x_1$? In two decimal places.
Just click on the link below for my working, I think I got it wrong. Can someone help m... | let $x$ be a fixed point, then $f(x)=x$, solve this equation.
Or you can think of a function $g(x)=f(x)-x$ and find the roots
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving a positive continuous function on $\mathbb{R}$ with $\lim\limits_{x\rightarrow\pm\infty}f(x) = 0$ has a maximum, with somewhat of a twist Suppose that $f(x)$ is continuous and $>0$ on $ I = \mathbb{R}$, and $\lim\limits_{x\rightarrow \pm \infty}$f(x) = 0.
(a) Prove $f(x)$ has a maximum on $I$.
For this, I gave ... | For point (b) a weaker hypothesis is that $f >0$ on $D\subset I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How many subsets of squares in a $3 \times 3$ grid, corners requiring both adjacent squares to be included? Given that I have a $3 \times 3$ grid of squares:
$$
\begin{matrix}
A & B & C \\
D & E & F \\
G & H & I \\
\end{matrix}
$$
I want to know: what are all the possible subsets of this set of squa... | Simply order the subsets by the number of corner squares they include. Using the symmetries of the square leaves very little work to be done.
0.The sets with no corner squares are the subsets of $\{B,D,F,H\}$, of which there are $16$.
*
*For each corner square, there are precisely four sets containing that corner an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Where can I validate $300\ 000$ digit prime number is valid one? I recently found a different method to compute prime number in $\mathcal O(\log(\log n))$ complexity. At present, that logic working fine for $300$ digits prime number, which I found on websites.I need to validate whether that logic will be working fine f... | From what you write I understand that you want to prove that your method is fine. It probably is if you checked up to 300, digits. But the only way to validate a method is by analysing it step by step and actually prove that it works. No matter how many digits you try, that will not be a proof that your method has no f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2721836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Given X that is unif(0,1), find the pdf of Y=1/sqrt(X) I have this problem which I would like to discuss whether my solution is accurate. There are no solved examples like this one in my text book, so I want to ask anyone if they think my solution is correct or not.
"Let $X$~ $unif(0,1)$. Find the pdfs of the followin... | Hint: if $X$ takes values between $0$ and $1$, where does $Y$ take values? You're correct that the pdf of $Y$ is $2 y^{-3}$ for some set of $y$, but it's not for all $y > 0$. Another way to think about this: for what $x$ is your statement $$1 - F_X(1/x^2) = 1 - 1/x^2$$
valid?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $F_n$ is the Fibonacci sequence, show that $F_n < \left(\frac 74\right)^n$ for $n\geq 1.$ Recall that the Fibonacci sequence is defined by $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n−1} + F_{n−2}$ for $n ≥ 2$. Prove that:
$$\forall \,\, n ≥ 1 ,\,\, F_n < \left(\frac 74\right)^n$$
In this question I understand how to do t... | The proposition that you're trying to prove is that $F_n<(\frac{7}{4})^n$
For $n = 0$, this is trivial; $0 < (\frac{7}{4})^0$
For $n = 1$, we have $1 < (\frac{7}{4})^1$
For your induction step, you assume that for all k < n, $F_k<(\frac{7}{4})^k$
So $F_{n-2}<(\frac{7}{4})^{n-2}$ and $F_{n-1}<(\frac{7}{4})^{n-1}$
$F_{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Error in solution to differential equation? I'm trying to solve $w^5L\frac{dL}{dw} = 1 - w^4L^2$. I attempted the substitution $M = \frac12w^5L^2$, so that $M' = \frac52w^4L^2 + w^5L'L$. Then
$$
w^5L'L = M' - \frac52 w^4L^2 = M' - \frac{5M}w \\
1 - w^4L^2 = 1 - \frac{2M}w
$$
so the equation becomes
$$
M' = 1 + \frac{3M... | I checked everything and everything was ok except here
$$M = c_1w^3 - \frac12w$$
$$w^5L^2 = 2c_1w^3 -w$$
$$w^2L^2 = 2c_1 -\frac 1 {w^2}$$
$$\pm Lw= \sqrt{2c_1 -\frac 1 {w^2}}$$
$$ L= \pm \frac 1 {w^2}\sqrt{2c_1w^2 -1}$$
$$ \boxed{L= \pm \frac {\sqrt{cw^2 -1}}{w^2}}$$
It's more simple to substitute $S=L^2$
$$w^5L\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
What "real numbers" (elements in $\mathbb R$), are people referring to? To define mathematical objects, it seems one defines them in terms of other mathematical objects.
However various mathematical objects have different "definitions".
E.g. it seems people "construct" the real numbers (they use objects other than the ... | I'd go for (1), slightly modified. If you've just constructed the real numbers in some book or paper, using, say, Dedekind cuts, then in that context a real number is a Dedekind cut, and you use properties of Dedekind cuts to prove theorems about real numbers.
Since there is a unique ordered field (up to unique isomorp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$F^n$ as a direct sum of cyclic submodules Let $A$ be an $n\times n$ matrix over a field $F$. Denote by the same letter $A$ the linear operator $F^n\to F^n$ given by $X\mapsto AX$. Endow $F^n$ with the structure of an $F[t]$-module by defining scalar multiplication as follows: if $f(t)\in F[t]$ and $X\in F^n$, then $f(... | The key result is that we have an isomorphism of $F[X]$-modules $F^n\simeq F[X]^n/\ker(XI_n-A)$.
I don't have much time now, so I leave you to find some references for now on the web or in the standard books. If I have time tonight (French time) , I will edit my answer and provide a full proof.
Now, you just apply the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.