Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Can $f(g(x))$ be a polynomial? Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions.
Is it possible that $f(g(x))$ is equal to a polynomial ?
edit
Some comments :
I was thinking about iterations.
So for instance $f(f(x)) = $ some polynomial.
However such $f$ are usually (Always ?) not entire because of the fac... | There are various ways to show that this can not be true. E.g., by Casorati-Weierstrass, the image of $|z|>R$ under $g$ is dense in the plane for every $R>0$, so the image of the same domain under $f\circ g$ contains a dense subset of $f(\mathbb{C})$ which is itself dense in the plane, showing that $f \circ g$ has an e... | {
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"timestamp": "2023-03-29T00:00:00",
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What happens if we change the definition of quotient ring to the one that does not have ideal restriction? From Wikipedia:
Given a ring R and a two-sided ideal I in R, we may define an
equivalence relation ~ on R as follows:
a ~ b if and only if a − b is in I.
Using the ideal properties, it is not difficult to chec... | No. You can't take just any subset $I$ of $R.$ To define quotient ring, first it required to be a group. So the subset $I$ has to be a subgroup of $(R, +).$ (Actually we require normal subgroup to define quotient group, but in this case $(R,+)$ is abelain. So any subgroup is normal.) Now take $I$ to be a subgroup of $(... | {
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How is the exponential in the Fourier transform pulled out of the integrand? I'm looking at Fourier Transforms in a Quantum Physics sense, and it's useful to associate the Fourier Series with the Dirac Delta. The book I'm using follows this argument (Shankar, Quantum Mechanics):
The Dirac Delta has the following proper... | The steps missing:
$$\begin{align}f(x^{\prime})&=\int_{-\infty}^{\infty}e^{jkx^{\prime}} \left(\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-jkx}f(x) \,\,\mathrm{d} x \right) \,\,\mathrm dk\tag{0}\\
&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{jk(x'-x)}f(x)\,dx\,dk\tag{1}\\
&=\int_{-\infty}^{\infty}\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115869",
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Why is $\cos((\omega+\alpha\cos(\omega' t))t)$ the wrong model for frequency modulation? So I was trying to program vibrato, or freqency modulation, naively using the model:
$$\cos((\omega + \alpha\cos(\omega' t))t)$$
Where $\alpha \lt \omega$ and $\omega' \ll \omega$. For practical purposes, assume $\omega=200\,\math... | Let $\theta(t) = \omega t + t \alpha \cos (\xi t)$, then
$\theta'(t) = \omega + \alpha \cos (\xi t) - t \alpha \xi \sin (\xi t)$, so you
can see that the instantaneous frequency is unbounded.
| {
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Soft question: Union of infinitely many closed sets this is a question that is not addressed in my book directly but I was curious. We just proved that the union of a finite collection of closed sets is also closed, but I was curious about if the union of infinitely many closed sets can be open. This question may not b... | In $\Bbb R$ take $\aleph_0$ closed intervals from $-n$ to $n$ for natural $n$: $$\bigcup_{n\in\Bbb N} [-n,n] = \Bbb R$$
Similary $$\bigcup_{n\in\Bbb N^+} \left([-n,-n+1] \cup [n-1,n]\right) = \Bbb R$$
| {
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Colimit preserves monomorphisms under certain conditions I know that colimit preserves epimorphisms.
Consider the special case where
*
*The diagrams are indexed by a directed set $I$,
*We are in the category of certain algebraic structures, such as $\mathbf{Ab}, \mathbf{Ring}, \mathbf{Set}$, $\mathbf{Mod_R}$, w... | Good observation!
Indeed it happens often that filtered colimits preserve finite limits, and as for a morphism $f: X\to Y$ being mono is equivalent to $id, id: X\rightrightarrows X$ being a pullback of $(f,f)$, in these situations monomorphisms are preserved.
You find this imposed as a condition in topos theory, wher... | {
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Rigorous proof for a maximization problem Problem: Eight players entered a round-robin tennis tournament. At the end of the tournament, a player who wins $N$ sets will take home $N^2$ dollars. The entry fee is $17.50 per player. Why is this enough to pay for the prizes?
Intuitive solution: Trying to maximize the total ... | Denote by $x_k$ the number of victories for player $P_k$, and assume $x_1\leq x_2\leq\ldots\leq x_8$. If $x_8<7$ then $P_8$ has lost at least one game against a player $P_k$ with $k<8$. Since
$$(x_8+1)^2+(x_k-1)^2=x_8^2+x_k^2+2(x_8-x_k)+2\geq x_8^2+x_k^2+2\ ,$$
the total payout would have been greater when $P_8$ would ... | {
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What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used. What is the remainder when $x^7-12x^5+23x-132$ is divided by $2x-1$? (Hint: Long division need not be used.)
The Hint is confusing!
| You can use Horner schema to evaluate it. For an explanation of this algorithm read this. In your case you have
\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline &1&0&-12&0&0&0&23&-132 \\\hline\frac12 &0&\frac12&\frac14&-\frac{47}{8}&-\frac{47}{16}&-\frac{47}{32}&-\frac{47}{64}&\frac{1425}{128}\\\hline &1&\frac12&-\frac{47}{4}&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1116272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Series proof $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$ Iam stuck with this proof. There seems to be no property to help.
If $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$ and that the reverse isnt true.
| $\sum_1^\infty{a_n^2} \leq (\sum_1^\infty|a_n|)(\sum_1^\infty|a_n|) < \infty$
A example to show reverse untrue.
See $a_n = \frac {1}{n} $
| {
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Prove that this sequence of integers is on average equal to zero. Consider the sequence $\{a(n)\}_{n\in\mathbb{N}^*}$ that is defined by the Dirichlet series:
$$\zeta (s)^2\cdot\left(1-\frac{1}{2^{s-1}}-\frac{1}{3^{s-1}}+\frac{1}{6^{s-1}}\right)=\sum_{n\geq 1}\frac{a(n)}{n^s}$$
Prove or disprove that on average it is e... | We have:
$$ \zeta(s)=\prod_{p}\left(1-\frac{1}{p^s}\right)^{-1},\qquad \zeta(s)^2=\sum_{n\geq 1}\frac{d(n)}{n^s} $$
hence, assuming that the sequence $\{a(n)\}_{n\in\mathbb{N}^*}$ is the sequence of coefficients of the Dirichlet series associated with $f(s)$:
$$ f(s)=\zeta(s)^2\left(1-\frac{1}{2^{s-1}}\right)\left(1-\f... | {
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Rotating a point in space about another via quaternion I have a system that is giving me a point in 3D space (call it (x, y, z)) and a quaternion (call it (qw, qx, qy, qz)). I want to create a point at (x+1, y, z), and then rotate that point using the quaternion. How would I do this?
The specific application that I a... | The answer to this question starts with the answer in this thread. Specifically, the formula posted as P' = Q(P-G)Q'+G, where P is the coordinates of the point being rotated, G is the point around which P is being rotated, Q is the quaternion, Q' is the quaternion inverse, and P' is the new location of the point after... | {
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Polynomial prove exercise $P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have to solve it for me, I'd be just happy to get a clue how - + if someone could provide me with ... | Since $a_j\geq 0$ $(1\leq j\leq n-1)$, all zeros are negative. Hence
$$ P(x)=(x+r_1)\cdots (x+r_n)\qquad r_j>0, 1\leq j\leq n$$
and $r_1\cdots r_n=1$. It follows from this last equality that at least one $r_j$ is greater than or equal to $1$, which gives $r_1+\cdots+r_n\geq 1$.
It follows that
$$ P(2)=(2+r_1)\cdots (2+... | {
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How do you show that the degree of an irreducible polynomial over the reals is either one or two? The degree of irreducible polynomials over the reals is either one or two.
Is it possible to prove it without using complex numbers? Or without using fundamental theorem of algebra?
| 1) If you know that every irreducible polynomial over $\mathbb R$ has degree $1$ or $2$, you immediately conclude that $\mathbb C$ is algebraically closed:
Else there would exist a simple algebraic extension $\mathbb C\subsetneq K=\mathbb C(a)$ with $[K/\mathbb C]=\operatorname {deg}_\mathbb C a=d\gt 1$.
Then $K=\m... | {
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Finding all possible combination **patterns** - as opposed to all possible combinations I start off with trying to find the number of possible combinations for a 5x5 grid (25 spaces), where each space could be a color from 1-4 (so 1, 2, 3, or 4)
I do 4^25 = 1,125,899,906,842,624 different combinations
However, now I'm ... | $\sum _{n=1}^c \mathcal{S}_{x y}^{(n)}$ is what you're after, with $x, y, c$ the two dimensions and the number of colors. $\mathcal{S}_{x y}^{(n)}$ is the Stirling number of the second kind.
This generalizes to arbitrary dimensions.
| {
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What class would this be covered in? Would the material in this section of a wikipedia article be covered in a standard course on Differential Geometry, or should I look elsewhere to learn those sorts of things? Specifically, topics like solid angles, line/surface/volume elements, and so forth. I have taken vector calc... | A standard differential geometry class would certainly contain this material, but it would probably not spend significant time on them. This is more the flavor of a multivariable calculus/analysis class; if this wasn't covered to your satisfaction in yours, you might want to study these on your own.
| {
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Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate? Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$.
Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate f... | The same idea also gives $p_n\leq 4^n$ and a rather trivial improvement of that is:
$n\in\Bbb Z_+,\;1\leq x\leq p_n\implies x=p_1^{e_1}\cdots p_n^{e_n}\cdot m^2,\;e_i=0,1\wedge\;m^2<p_n$.
If $e_n=1$ then $0=e_1=\cdots=e_{n-1}$ and if $e_n=0$ there is at most $2^{n-1}$ ways to chose $x$. Similar to Erdős proof $p_n\leq(... | {
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Given a graph on $n$ vertices find the maximum amount of edges so it can be colored with no monochromatic $K_m$ I invented a problem and I wanted to share :What is the maximum amount of edges a graph on $n$ vertices can have if it can be edge-colored with $k$ colors so that it does not have a monochromatic $K_m$?
I th... | Such a graph cannot contain a complete subgraph of size $R\underbrace{(m,m\dots m)}_{\text{k times}}=j$ . What is the subgraph on $n$ vertices that has the most edges and does not contain a complete sugraph of size $n$? It is the Turan graph $T(n,j-1)$, so if $T(n,j-1)$ works we are done.
By the definition of the Ramse... | {
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How to understand the concept behind the equation $\boldsymbol{Ax}=\boldsymbol{b}$ As is know to all, the equation $\boldsymbol{Ax}=\boldsymbol{b}$ can be understand as to find the linear combination coefficient of the column vector of the matrix $A$. At the same time, it can also be explained as to find vector space w... | As far as the equation $Ax=b$ goes, the left-hand-side can the thought of in different ways.
One way is that if $a_1,\ldots,a_n$ are the columns of $A$, and $x=(x_1,\ldots,x_n)$, then
$Ax = a_1x_1+\cdots+a_nx_n,$
which is a linear combination of the columns of $A$. Thus to solve the equation, you need to find the value... | {
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class equation of order $10$ Is it a class equation of order $10$
$10=1+1+1+2+5$.
As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is the case. So I think it is a valid class equation.Is it correct?
| This cannot be a class equation of a group of order 10 because the $1's$ in the class equation correspond to all those elements which lie in the center $Z(G)$ of the group. But $Z(G)$ is a subgroup, so by Lagrange it's order (which in this case will be $3$) must divide $|G|=10$. This is a contradiction.
| {
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Expectation value of absolute value of difference of two random variables I do not really know how to prove the following statement:
If $E(|X-Y|)=0$ then $P(X=Y)=1$.
The main problem is how to handle the absolute value $|X-Y|$.
If I say that $|X-Y| \geq 0$ it follows that $E(X)=E(Y)$ which is
also the result for $... | Simple proof by contradiction:
Assume that $P(X=Y)<1$, then there has to exists $x',y'$ such that $x'\neq y'$ and $P(X=x'\wedge Y=y')>0$.
But then if we look at the random variable $Z=|X-Y|$, we get that
$$E(Z)=\sum_{x,y}P(X=x\wedge Y=y)\cdot|x-y|$$
$$=P(X=x'\wedge Y=y')\cdot|x'-y'|+\sum_{x,y\neq(x',y')}P(X=x\wedge Y=y... | {
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Differential equation degree doubt $$\frac{dy}{dx} = \sin^{-1} (y)$$ The above equation is a form of $\frac{dy}{dx} = f(y)$, so degree should be $1$. But if I write it as $$y = \sin\left(\frac{dy}{dx}\right)$$ then degree is not defined as it is not a polynomial in $\frac{dy}{dx}$. Please explain?
| To make sense of the definition of the order of a differential equation, you need to first isolate the highest order derivative first. For example,
$\frac{dy}{dx} = y, \frac{dy}{dx} = y^2 + 1$ are of the first order.
$ \frac{d^2 y}{dx^2} = y, \frac{d^2 y}{dx^2} = (\frac{dy}{dx})^3 + 1$ are of second order.
Your diffe... | {
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Building a proper homomorphism between groups. Suppose I have a cyclic group $G$ of order $6$. I want to show that it is isomorphic to $\Bbb {Z}_6$. So $G=\{e,g^2,g^3,g^4,g^5\}=\langle g\rangle$. Can I build a homomorphism $f:G \to \Bbb{Z}_6$ that way?
$f(x)=f(g^m)=mf(g)=m$ where $f(g)=1$ and $f(e)=0$. It is problemati... | It is much better to build the isomorphism in the other direction, as
$$
\varphi([i]) = g^{i},
$$
if $[i]$ is the class of $i$ in $\mathbb{Z}_{6}$. You have to prove it is well-defined, but this is immediate.
Even better, start with the homomorphism
$$
\mathbb{Z} \to G, \qquad i \mapsto g^{i},
$$
and use the first isom... | {
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Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$ Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$
What is the smallest possible value of $abcd$?
I just need a few hints, nothing else. How should I begin? Number theory?
| The number of representations of a positive integer as a sum of two squares depends on the number of prime divisors of the form $4k+1$ (see Cox, Primes of the form $a^2+k b^2$). If we take the first two primes of such a form, $5$ and $13$, we have that $5\cdot 13$ can be represented as:
$$ 65 = 1^2+8^2 = 4^2+7^2 $$
so ... | {
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How do I solve this Olympiad question with floor functions?
Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part equals 1 for the first time. What is the difference between the... | Clearly we must have $x_3<4$, $x_2<16$, and $x_1<256$, so the largest possible value of $x_1$ is $255$. Since $x_3>1$, we know that $x_3\ge 2$ and hence that $x_2\ge 4$. This implies that $x_1\ge 16$. Thus, the range is from $16$ through $255$, the difference being $239$.
| {
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Let $F$ be a linear operator such that $F^2 - F + I = 0$, show that $F$ is invertible and $F^{-1} = I - F$ I didn't understand this exercise. I tried working with
$$F^2 - F + I = 0\implies (F-I)(F) + I =0$$
but I really don't understand how to prove $F$ is invertible neither find the inverse. Any hints? Thank you so mu... | Hint 1: can $0$ be an eigenvalue of $F$?
Hint 2: can you multiply $F^2-F+I=0$ by $F^{-1}$ once you know it exists?
| {
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Find the number of children, given that the estate was divided evenly between them Problem of the Week at University of Waterloo:
A man died leaving some money in his estate. All of this money was to be
divided among his children in the following manner:
*
*$x$ to the first born plus $1/16$ of what remains, then... | Hint
Let $y$ be the amount he left over. How much did the first born receive? How much did the second born receive?
Mathe them equal.
| {
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Order of any element divides the largest order. Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect in it, but I can't figure out what....Also I didn't use that fact that $A... | Let $x\in A$ such that $x$ has largest order $m$ in $A$. Assume that $g\in A$ such that $|g|$ does not divide $m$. Wlog, assume that $|g|=p$ where $p$ is a prime. Then $gcd(p,m)=1$. Now, as $A$ is abelian, $g$ and $x$ commute and $|gx|=pm=lcm(p,m)$. Note that $lcm(p,m)=pm$ as $p$ is a prime and $p\nmid m$. Hence, $|gx|... | {
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Can the cube of every perfect number be written as the sum of three cubes? I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is
$$x^3+y^3+z^3=\sigma^3$$
where $\sigma$ is a perfect number
(well it holds good for the first three perfect numbe... | This is more of a comment as opposed to an answer
There is a formula for finding the values of $a, b, c, d$ in the following equation: $$a^3 + b^3 + c^3 = d^3$$ Where $$\forall x, y\in \mathbb{Z}, \ \begin{align} a &= 3x^2 + 5y(x - y), \ b = 2\big(2x(x - y) + 3y^2\big) \\ c &= 5x(x - y) - 3y^2, \ d = 2\big(3x^2 - 2y(x... | {
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"answer_id": 1
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What does 'forms a right-handed set' mean? In a question I am reading, the following question appears.
What if $\vec{A},\vec{B}$, and $\vec{C}$ are mutually perpendicular and form a right-handed set?
What exactly does "form a right-handed set" mean? That $\hat{A}\times\hat{B}=\hat{C}$? Or does it mean the stronger eq... | There are several equivalent definitions, but here is one that is often convenient:
The ordered triplet $(\vec A,\vec B,\vec C)$ of vectors in $\mathbb R^3$ is right called right-handed if $\det(\vec A,\vec B,\vec C)>0$, where $(\vec A,\vec B,\vec C)$ is the square matrix formed by taking the three vectors as columns.
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partition of fractional parts where each sum of them has to be at least 1 Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace $ stands for the fractional part of $ x $.
Ar... | No. Let $a_i=0.99$ for all $i$ and $t=6$. Then $k=5$. But if $ A_j \subseteq \lbrace 1,\ldots,t \rbrace $ with $ \sum_{i \in A_j} \lbrace a_i \rbrace \ge 1 $ then $|A_j|\ge 2$, so there can be at most $3<4=k-2$ such mutually disjiont subsets.
PS. Instead of fractional parts we can simply consider the rational numbers $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve second order congruence equation if modulo is not a prime number the equation is $x^2 = 57 \pmod{64}$
I know how to solve equations like
(*) $ax^2 +bx +c = 0 \pmod{p}$, where $p$ is prime
and i know all the definitions for like Legendre's Symbol and all of the other quadratic residue terminology and that... | First, also from the reciprocity law, we know that $\;x^2=z\pmod{2^k}\;$ , has a solution for $\;k\ge 3\;$ iff $\;z=1\pmod 8\;$, which is the case here.
Now:
$$x_k^2=z\pmod{2^k}\implies x_{k+1}=\begin{cases}x_k+2^{k-1}&,\;\;\frac{x_k^2-z}{2^k}=1\pmod 2\\{}\\x_k&,\;\;\frac{x_k-z}{2^k}=0\pmod 2\end{cases}$$
and you can c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An easy example of a non-constructive proof without an obvious "fix"? I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it.
Euclid's proof of the infinitude of primes came to mind, however there is ... | There's a classic example from point-set topology. The statement is this: suppose $X$ is a topological space and $Y \subseteq X$ is such that for all $y \in Y$ there exists an open set $U$ such that $y \in U$ and $U \subseteq Y$. Then $Y$ is open.
Most introductory textbooks will tacitly use the axiom of choice along t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1118810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "57",
"answer_count": 15,
"answer_id": 14
} |
Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$
Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$
Why is it wrong to...
$$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
| You forgot to find the derivative of $e^x -1$ to replace $dx$ in the original integral.
Put $e^x - 1 = u$. Then $e^x = u+1$.
Then $du = e^x\,dx\iff dx = \frac{du}{u+1}$
This gives us $$\int_0^1 \frac{u^{1/2}}{u+1} \,du$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove the identity $\prod_{n=1}^{\infty} (1-q^{2n-1}) (1+q^{n}) =1 $ for $|q|<1$? Eulers product identity is as follows
\begin{align}
\prod_{n=1}^{\infty} (1-q^{2n-1}) (1+q^{n}) =1
\end{align}
How one can explicitly prove this identity?
Note here $q$ deonotes a complex number satisfying $|q|<1$
| For $\lvert q\rvert < 1$, define
$$f(q) = \prod_{n=1}^\infty (1-q^{2n-1})(1+q^n).$$
The product is absolutely and locally uniformly convergent on the open unit disk, hence $f$ is holomorphic there, and the product can be reordered and regrouped as desired. It is clear that $f(0) = 1$, and by reordering and regrouping, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about a notation Let $A$ be a non-singular square matrix. Which of the following notations is correct?
$${A^2}^{-1} \qquad \text{or} \qquad A^{-2}$$
| I assume that the first one is interpreted as $(A^2)^{-1}$. Note that $A^{-2}$ is defined as $(A^{-1})^2$. Hence, if you want to interpret the two as meaning the same thing you have to think about it:
Exercise: Determine whether or not $(A^2)^{-1}=(A^{-1})^2$ for all non-singular square matrix $A$.
| {
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showing a (B,p) is complete let $(A,d)$ be a metric space which is complete and let $B$ be closed in A. Then prove $(B,p)$ is complete, with $p = d|_{B\times B}$ (i.e. restriction of d onto $B\times B$)
thoughts:
since $B$ is closed in $A$ and $(A,d)$ is complete i figured that $(B,y)$ is complete - from here I know I ... | Let $X$ be a complete metric space and $F$ a closed set in $X$. If $x_n$ is a Cauchy sequence in $F$, then it has a limit $x\in X$. Since $F$ is closed, $x\in F$. Hence $F$ is complete.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does the dot product angle formula work for $\Bbb{R}^n$? Whenever I have seen this formula discussed
\begin{equation}
\textbf{A} \cdot \textbf{B} = \|\textbf{A} \| \|\textbf{B} \| \cos\theta
\end{equation}
I have always seen it using vectors in $\Bbb{R}^2$.
I was wondering if this property works if $\dim \textbf{A}= ... | Yes it does. If you accept that this holds in $\mathbb R^2$, the fact that it holds in $\mathbb R^n$ follows fairly easily - notice that for any choice of two vectors $A$ and $B$, it is always possible to choose a two-dimensional subspace (i.e. a plane) containing both vectors (the span of the two vectors usually funct... | {
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"timestamp": "2023-03-29T00:00:00",
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Elliptic differential operator I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are elliptic. Does anybody know how to do this, I am really puzzled by the wikipedia definition of ... | The heat equation is parabolic:
$$
\frac{\partial f}{\partial t} = \frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}
$$
Laplace's equation is elliptic:
$$
\frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\par... | {
"language": "en",
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Tensoring the exact sequence by a faithfully flat module I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves".
First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ if it satisfies one of the following three equivalent conditions... | Although this is the usual property that defines the faithfully flatness, (one of the) Liu's definition(s) which is helpful here is the following:
Let $M$ be a flat $A$-module. Then $M$ is faithfully flat iff for any $A$-module $N$ such that $M\otimes_AN=0$ we have $N=0$.
Recall that $N'\stackrel{u}\to N\stackrel{v}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Universal Cover of $\mathbb{R}P^{2}$ minus a point I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can show it is homotopy equivalent to the boundary which is homotopy equiva... | Letting $M$ denote the Mobius band, there is a universal covering map $\mathbb{R}^2 \to M$ with infinite cyclic deck transformation group $\langle r \rangle$ generated by the "glide reflection"
$$r : (x,y) \to (x+1,-y)
$$
As with any universal covering map, $M$ is the orbit space of this action. One can verify this by ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.
If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.
I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure and that a countable set has measure zero.
Remark: $\mu(\... | Note that if $S \subset T$, then $\mu(S) \leq \mu(T)$ provided both $S$ and $T$ are measurable. So, $\mu(A) \leq \mu(A \cup E)$, and you already have the other inequality.
Similar idea will work for the next equality as well.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Moment generating function gives an undefined first moment, but first moment still exists? Let's say we have a probability density function given by
$f_X(x) = 2x$ for $0 \leq x \leq 1$.
(Note $\int_0^1 f_X(x) = 1$.)
The moment generating function is
$$\int_0^1 e^{tx}\cdot2x \,dx$$
which wolfram alpha gives as
$$\frac{2... | The singularity of the MGF at $t=0$ is removable. Use the Maclaurin series of the exponential function. The MGF is actually defined everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1119803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On average, how many times must I roll a dice until I get a $6$? On average, how many times must I roll a dice until I get a $6$?
I got this question from a book called Fifty Challenging Problems in Probability.
The answer is $6$, and I understand the solution the book has given me. However, I want to know why the fol... | The probability of the time of first success is given by the Geometric distribution.
The distribution formula is:
$$P(X=k) = pq^{n-1}$$
where $q=1-p$.
It's very simple to explain this formula. Let's assume that we consider as a success getting a 6 rolling a dice. Then the probability of getting a success at the first t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1119872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
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When is a curve parametrizable? Is there a way in general to tell whether a given curve is parametrizable?
|
Suppose you are considering the level set $$f(x,y)=c$$ and we want to study whether we can parametrize this curve near the point $(x_0,y_0)$, which is a point in it, $f(x_0,y_0)=c$. If one of the partial derivatives is non-zero at that point, say $\frac{\partial f}{\partial y}(x_0,y_0)\neq0$ then we can use the Implic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Poincare duality isomorphism problem in the book "characteristic classes" This is the problem from the book, "characteristic classes" written by J.W. Milnor.
[Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k = p-n$.
Show that the Poincare d... | Start picking in the upper left angle the element $u' \otimes \mu_A \in H^k(A,A\setminus M)\otimes H_p(A)$ (using Milnor's notation $u'$ is the dual cohomology class of $M$), then you have to chase this element a little bit. The left "path" $( \downarrow$ and then $\rightarrow)$ is straightforward. You end up with $u'|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Decide the smooth function $r : \mathbb R \rightarrow \mathbb R$ of the equation $r(t)^2 + r'(t)^2 = 1$. Suppose $r:\mathbb R \rightarrow \mathbb R$ is a smooth function and suppose $r(t)^2 + r'(t)^2 = 1$.
I want to determine the function $r(t)$.
I see that $r(t)^2 + r'(t)^2 = 1$, so I could take $r(t) = \pm \cos(t), \... | This is one of the rare occasions where it actually helps to differentiate the given ODE! In this way we obtain
$$2r'(t)\bigl(r(t)+r''(t)\bigr)=0\qquad\forall t\ .\tag{1}$$
This is satisfied when $r'(t)\equiv0$, or $r(t)=c$ for some $c\in{\mathbb R}$. From the original ODE it then follows that $c^2=1$, so that we obtai... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine significant figures involving radicals and exponents How do you determine the significant figures for solving equations with radicals and exponents? For example, how do you evaluate $x = \sqrt{4.56^2 +1.23^2}$?
| Here is an abbreviation of the explanation I use in my 11th grade Chemistry and 12th grade Physics classes. This uses precision as is often used in American secondary schools, though it is not usually explained in quite this way.
There are two ways to measure precision: significant figures and decimal places. Significa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Use integration by substitution I'm trying to evaluate integrals using substitution. I had
$$\int (x+1)(3x+1)^9 dx$$
My solution: Let $u=3x+1$ then $du/dx=3$
$$u=3x+1 \implies 3x=u-1 \implies x=\frac{1}{3}(u-1) \implies x+1=\frac{1}{3}(u+2) $$
Now I get $$\frac{1}{3} \int (x+1)(3x+1)^9 (3 \,dx) = \frac{1}{3} \int \fr... | $\frac{19683 x^{11}}{11}+\frac{39366 x^{10}}{5}+15309 x^9+17496 x^8+13122 x^7+6804 x^6+\frac{12474 x^5}{5}+648 x^4+117 x^3+14 x^2+x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Is $\text{Hom}(\prod_p \Bbb Z/p\Bbb Z, \Bbb Q) = 0$ possible without choice? That divisible abelian groups are precisely the injective groups is equivalent to choice; indeed, there are some models of ZF with no injective groups at all. Now, given that $\Bbb Q$ is injective, one immediately has that $\text{Hom}(\prod_p ... | A related question is the following: does there exist a model of ZF where choice fails but the hom-set is nontrivial? The following is an answer by Andreas Blass.
The answer to that is yes. The reason is that the failure of choice can be such as to involve only very large sets, much larger than the groups in the ques... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Convergence of Uniformly Distributed Random Variables (n-dimensional) Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ${Q_n}(dx) = {2^{-n}} {{\chi}_{C_{n}(x)}}{m_n}(dx)$, wher... | A way to show convergence in probability to $0$ is to show that $\mathbb E\left[\sqrt{X_n}\right]\to 0$. This can be done in the following way: we observe that
$$
\mathbb E\left[\sqrt{X_n}\right]=2^{-n}\int_{[0,2]^n}\prod_{i=1}^n\sqrt{x_i}\mathrm dx_1\dots\mathrm dx_n=\left(\frac 12\int_0^2\sqrt u\mathrm du\right)^n
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $2^{3^n} + 1$ is divisible by 9, for $n\ge1$
Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.
Work of OP: The thing is I have no idea, everything I tried ended up on nothing.
Third party commentary: Standard ideas to attack such problems include induction and congruence arithmetic. (The answe... | Hint $\rm \,\ {\rm mod}\,\ A^{\large B} + 1\!:\,\ \color{#c00}{A^B\equiv -1}\ \Rightarrow\ \color{}{A}^{\large BC}\equiv (\color{#c00}{A^{\large B}})^{\large C}\equiv (\color{#c00}{-1})^{\large C} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$ I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$
I did prove convergence:
$0<\theta<1$
$$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$
Now I want... | We can follow the proof in the post indicated by Marko Riedel. Rewrite $$\underset{n=1}{\overset{\infty}{\sum}}\frac{\sin\left(nx\right)}{n^{3}}=x^{3}\underset{n=1}{\overset{\infty}{\sum}}\frac{\sin\left(nx\right)}{\left(nx\right)^{3}}$$
and use the fact that the Mellin transform identity for harmonic sums with base f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How can one know every Cauchy sequence in a complete metric space converges? I am new to Cauchy sequences. I stumbled onto them in the process of learning what a Hilbert space is. As I understand it, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
But ... | As I see it, the question is, "How can the criterion in the definition of completeness ever be verified (in principle, to say nothing of in practice), since the definition makes an assertion about arbitrary Cauchy sequences, and most spaces admit A Lot of Cauchy sequences?"
If one examines the proofs for, say, compact ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to prove $L_{f}(P) \leq L_{f}(Q)$ when $Q$ and $P$ are partitions of $[a,b]$ and $Q \supseteq P$ I'm having trouble proving this idea.
Suppose that $f$ is bounded on the interval $[a, b]$. $P$ and $Q$ are partitions of $[a, b]$, and $Q \supseteq P$.
$$
L_{f}(P) \leq L_{f}(Q)
$$
I know that this makes sense, because... | First, note we can simply prove this for $P$ and $Q$ where $Q$ is obtained from $P$ by adding only one number $t^*$, and then go by induction on the number of extra points $Q$ has.
So suppose $P=\{t_0,\ldots,t_i,t_{i+1},\ldots,t_n\}$ and $Q=\{t_0,\ldots,t_i,t^*,t_{i+1},\ldots,t_n\}$. Let $m_i$ be the infimum of $f$ on... | {
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Is this a valid way to prove that $\frac{d}{dx}e^x=e^x$? $$e^x= 1+x/1!+x^2/2!+x^3/3!+x^4/4!\cdots$$
$$\frac{d}{dx}e^x= \frac{d}{dx}1+\frac{d}{dx}x+\frac{d}{dx}x^2/2!+\frac{d}{dx}x^3/3!+\frac{d}{dx}x^4/4!+\cdots$$
$$\frac{d}{dx}e^x=0+1+2x/2!+3x^2/3!+4x^3/4!\cdots$$
$$\frac{d}{dx}e^x= 1+x/1!+x^2/2!+\cdots=e^x$$
Of course... | It is valid after you have shown (or defined) that $e^x$ is indeed equal to that series and that deriving the series is the same as deriving each term individually (under suitable conditions, of course). It is also not circular and works also for complex numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Surprising applications of topology Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The idea is like this: consider $T = \{ (x,y,z) \mid x + y + z = 1, x, y, z \geq 0 \}$. ... | How about Furstenberg's proof of the infinitude of prime numbers?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 10,
"answer_id": 9
} |
Where can I find linear algebra described in a pointfree manner? Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the "eigenspace function of $A$" is the map $\mathrm{Eig}_A : \mathrm... | A commutative algebra textbook.
Here is a better description of how eigenspaces work from this perspective: a pair consisting of an $R$-module and an endomorphism of it is the same thing as an $R[X]$-module. If $R$ is a field, then finitely generated $R[X]$-modules are classified by the structure theorem for finitely ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Is there a notation for being "a finite subset of"? I would gladly use a notation for “$A$ is a finite subset of $B$”, like
$$A\sqsubset B \text{ or } A\underset{fin}{\subset} B,$$
but I have never seen a notation for that. Are there any?
EDIT: While waiting for a future standard, I will use Joffan’s $\ddot{\subset}... | $A\in[B]^{\lt\omega}$ where $[B]^{\lt\omega}$ denotes the set of all finite subsets of $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
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Does Intermediate Value Theorem $\rightarrow $ continuous? i try to understand Intermediate Value Theorem and wonder if the theorem works for the opposite side. I mean, if we know that $\forall c\:\:\:f\left(a\right)\le \:c\le \:f\left(b\right)\:,\:\exists x_0\in \left[a,b\right]\:\:$ such that $f\left(x_0\right)=c$ th... | No, what about $f(x)=x$ if $0\le x<1$ and $f(1)=1/2$. Clearly this has the property you stated with $a=0,b=1$ but is not continuous in $[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How do I find the error of nth iteration in Newton's Raphson's method without knowing the exact root In our calculus class, we were introduced to the numerical approximation of root by Newton Raphson method. The question was to calculate the root of a function up to nth decimal places.
Assuming that the function is nic... | This is a complement to Pp.'s answer.
Newton's method converges, and with the rate given in Pp.'s answer, only if certain assumptions are met. In simple numerical examples these assumptions are usually not tested in advance. Instead, one chooses a reasonable starting point $x_0$ and will then soon find out whether the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Check my answer - complex analysis, using residue and rouche's theorem I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this.
Question:
We are given $f(z)=2z-\sinh (z)$ defined on the circle $|z|=1$. Our goal is to calculate
$\in... | As @Mhenni Benghorbal said, you can do this a simpler way; that is, you don't need to use Rouche's Theorem. To find the poles, we only need to determine when
$$
2z - \sinh(z) = 0.
$$
Using the series representation of $\sinh(z) = \sum_{n=0}^{\infty}\frac{z^{2n+1}}{(2n+1)!}$, we have
$$
2z - \sum_{n=0}^{\infty}\frac{z^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Investigate the convergence of $\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}} \, dx$
Investigate the convergence of $$\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}} \, dx$$
so first of all let's split the integral to:
$$I_1 = \int_1^2 \frac{\cos x \ln x}{x\sqrt{x^2-1}} \, dx, \hspace{10mm} I_2 = \int_2^\infty \f... | For large enough $x$ you have $$\left| \frac{\cos x \ln x}{x^2} \right| \le \frac{\ln x}{2 x^2}.$$ You should have little difficulty showing that $$\int_1^\infty \frac{\ln x}{x^2} \, dx < \infty.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Geometry: What is the height of a solid pyramid So the question is OPQRS is a right pyramid whose base is a square of sides 12 cm each. Given that the the slant height of the pyramid is 15cm. And now I need to find the Height of the pyramid.
I did my equation like this:
QR 1/2 = 6cm
HEIGHT^2 = 6^2+12^2
... | The correct calculation is $\sqrt{15^2-(\frac{12}{2})^2}=\sqrt{189}=13,748$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Sum of nth roots of unity Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$
Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}{n}}}{1-e^{\frac{i2\pi}{n}}} =0$$ by use of the sum of geometric series'. My issue is proceedi... | If a finite set of complex numbers is symmetric about a line passing through the origin, then its sum must lie on that line; if it is symmetric about two different lines through the origin, then its sum must be zero. The $n^{\text{th}}$ roots of unity are the vertices of a regular $n$-gon centered at the origin, which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
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A relation between the Jacobson radicals of a ring and those of a certain quotient ring Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition (i.e. not using anything about s... | Lemma. Let $\phi : R\to S$ be an epimorphism. Then $\phi (J(R))\subset J(S)$:
proof. Let $x\in \phi (J(R))$. by Atiyah-Macdonald, Proposition 1.9, we need to prove that for all $s\in S$, $1-sx$ is a unit in $S$. let $x= \phi (t)$ for some $t\in J(R)$, and $s=\phi (r)$. we have $1-sx=\phi (1-rt)$. note that $1-rt$ has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Beginner Simplex problem Good evening,
I have started studying the simplex method for some examinations I would like to take, and to be perfectly honest, I am stuck really bad. The basic examples and exercises are simple, and I can generally solve them without problem. But once I get to the actual exercise part of the ... | And for house-keeping reasons, the answer as edited above is:
B c b P1 P2 P3 P4 P5 P6
P5 0 1 0 2 -1 0 1 0
P4 0 4 0 1/2 0 1 0 1/2
P1 -1 2 1 -2 1 0 0 -1
- z -2 0 0 2 0 0 1
The problem has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition.
proof:
given $a_n=a^{\frac{1}{n}}$ for $a\in\mathbb{R},a>1$.
for $n\in\mathbb{N}^*,n+1>n\Rightarrow \... | You didn't need the first paragraph in your proof, just the second paragraph. In the second paragraph, you cannot write $N = \frac{1}{\log_a(\epsilon + 1)}$ because $\frac{1}{\log_a(\epsilon + 1)}$ is not necessarily an integer. However, you can let $N > \frac{1}{\log_a(\epsilon + 1)}$. The rest of the proof is fine.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Stuck on an integration question… $$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$
The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$
but I keep going wrong.
Can someone show me how to get this solution?
Thanks.
| The form of the answer suggests integration by parts with the choice $$u = \cosh^{-1} \left( \frac{x}{2} + 1 \right), \quad dv = x^{-1/2} \, dx.$$ Then compute the derivative $$du = \ldots?$$ and the integral $$v = \ldots?$$ If you have trouble computing $du$, you can obtain it by writing $$x = \cosh u, \quad \frac{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For which $q\in\mathbb Q$ is $\sin(\frac\pi2q)$ rational?
Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q\;?$$
Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$?
As $\sin(\pi/6) = 1/2$ we have $q = 1/3$ is a solution. Are the... | The only rationals $r$ such that $\sin(\pi r)$ is rational are those for which $ \sin(\pi r)$ is in $\{-1,-1/2,0,1/2,1\}$. This is because $2 \sin(\pi r)$ is an algebraic integer, and algebraic integers that are rational are ordinary integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Probability of one stock price rising, given probabilities of several prices rising/falling So this is the problem:
An investor is monitoring stocks from Company A and Company B, which
each either increase or decrease each day. On a given day, suppose
that there is a probability of 0.38 that both stocks will incr... | The sum of the probabilities for all possible cases need to add to one.
Since the stocks must go up or down (not stay the same) there are four possible outcomes for two stocks: {$A\uparrow B \uparrow, A\uparrow B \downarrow, A\downarrow B \uparrow, A\downarrow B \downarrow $}.
If you're given three (disjoint) probabili... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Applying the Stone-Weierstrass Theorem to approximate even functions
Let $f:[-1,1] \rightarrow \mathbb{R}$ be any even continuous function on $[-1,1]$ (i.e. $f(-x)=f(x)$ $\forall x \in [-1,1]$). Let $\epsilon>0$. Prove that there exists an even polynomial $p$ such that $$|f(x)-p(x)|< \epsilon$$ $$\forall x \in [-1,1]$... | Hint: If $\sup_{x \in [-1,1]} |\,p(x) - f(x)|<\epsilon$ then
$\sup_{x \in [-1,1]} |\,\frac{1}{2}(\,p(x)+ p(-x)\,)- f(x)\,|<\epsilon$
$\bf{Added:}$
Let $p(x)$ a polynomial so that for every $x \in [-1,1]$ we have $|\,p(x) - f(x)|<\epsilon$. Note that if $x \in [-1,1]$ then also $-x \in [-1,1]$. So we have
$$|\,p(x) - f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is the integral of $\frac{\sqrt{x^2-49}}{x^3}$ I used trig substitution and got
$\displaystyle \int \dfrac{7\tan \theta}{343\sec ^3\theta}d\theta$
Then simplified to sin and cos functions, using U substitution with a final answer of:
$\dfrac{-7}{3x^3}+C$
Which section did I go wrong in. Any help would be apprecia... | If you let $x=7 \sec \theta$ then you won't get $7 \tan \theta$ in the numerator since $\sec^2\theta -\tan^2\theta =1$.
When the sub is done correctly you'll get an integral of the form $\displaystyle \frac{49}{343} \int \frac{\tan^2\theta}{\sec^2\theta}\,d\theta = \frac{1}{7} \int \sin^2\theta\ d\theta = \frac{1}{14}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Showing that $\mathcal{A}$ being countable $\Rightarrow f(\mathcal{A})$ is countable - (Algebras/Sigma Algebras)
For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at a loss o... | It is clear that $\sigma(\mathcal A)\subset\sigma(f(\mathcal A))$, as $\mathcal A\subset f(\mathcal A)$. For the reverse inclusion, $\sigma(A)$ is a $\sigma$-algebra that contains $f(\mathcal A)$, so as $\sigma(f(\mathcal A))$ is the intersection of all such $\sigma$-algebras, $\sigma(f(\mathcal A))\subset\sigma(A)$.
F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum $(1-x)^n$ $\sum_{r=1}^n$ $r$ $n\choose r$ $(\frac{x}{1-x})^r$ The question is to find the value of:
$n\choose 1$$x(1-x)^{n-1}$ +2.$n\choose2$$x^2(1-x)^{n-2}$ + 3$n\choose3$$x^3(1-x)^{n-3}$ .......n$n\choose n$$x^n$.
I wrote the general term and tried to sum it as:
S=$(1-x)^n$$\sum_{r=1}^n$$r$$n\choose r$$(\frac{x}{... | HINT: $r\binom{n}r=n\binom{n-1}{r-1}$; this is easy to see if you expand into factorials, and it also has a straightforward combinatorial proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?
In process of solving this problem, I faced to the problem of proving that
$A::$: $\dfrac{2k+1}{(k(k+1))^2}-\dfrac{4}{3^{k+1}}\geq 0$ for every positive odd integ... | Note first that $$\dfrac{2k+1}{(k(k+1))^2}\geqslant \dfrac{2k}{k^2(k+1)^2}= \dfrac{2}{k(k+1)^2}\geqslant \dfrac{2}{(k+1)^3}$$
If we prove that
$$\dfrac{1}{n^3}\geqslant \dfrac{2}{3^n} \tag{1}$$
which is equivalent to
$$\dfrac{3^n}{n^3} \geqslant 2 \tag{2}$$
then we can conclude that
$$\dfrac{2k+1}{(k(k+1))^2}-\dfrac{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123065",
"timestamp": "2023-03-29T00:00:00",
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Struggling with connection between Clifford Algebra (/GA) and their matrix generators As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of $\mathcal{Cl}_{3}(\mathbb C)$??).
But, I... | Those formulas for dot and wedge product are only valid when $A$ and $B$ are vectors, not if they are bivectors, trivectors, etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Divide matrix using left division In matlab, I defined
a=[1;2;3]
b=[4;5;6]
both a and b are not square matrix.
and execute a\b will return 2.2857
From various sources, we understand that a\b ~= inv(a)*b, but in my case, a is not a square matrix, thus we can't perform an inverse operation on it.
I am actually translati... | In Matlab $a $\ $b =(a'a)^{-1}a'b$, i.e. $b$ is projected onto $a$.
A little bit more information: Consider,
$$
b \approx ax
$$
where $x$ is unknown and there are no exact solutions i.e. $b$ and $a$ have many rows and few columns, then the typical way to solve this problem is to solve for $x$ by projecting $b$ onto $a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converge.
Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converges.
We cannot apply Abel's/Dirichliet's tests here (For example, Dirichliet's test demands that for $g(x)=\ln x$, $\int_0^1 g(x)dx < \infty$ which isn't true).
I also tried to compare the integral to another;
Sin... | Since:
$$ \int_{0}^{1}\frac{dt}{1-xt}=-\frac{\log(1-x)}{x} $$
and:
$$ I=\int_{0}^{1}\frac{\log x}{x-1}\,dx = -\int_{0}^{1}\frac{\log(1-x)}{x}\,dx $$
we have:
$$ I = \iint_{(0,1)^2}\frac{1}{1-xt}\,dt\,dx = \sum_{n\geq 0}\iint_{(0,1)^2}(xt)^n\,dt\,dx = \sum_{n\geq 0}\frac{1}{(n+1)^2}=\color{red}{\zeta(2)}$$
as wanted.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
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Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$. Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$.
What I did so far:
$f(x)=(x(2-x))^{-1}$. $f'(x)=-(x(2-x))^{-2}[2-2x]$ $f''(x)=2(x(2-x))^{-3}[2-2x]^2+2(x(2-x))^{-2}$. It confuses me a lot. I know I have to determine where I have an expression multiplied by $2-2x$ and wh... | Since for any $a\neq 0$:
$$\frac{1}{x(a-x)} = \frac{1}{a}\left(\frac{1}{x}+\frac{1}{a-x}\right)\tag{1}$$
we have:
$$ \frac{d^n}{dx^n}\frac{1}{x(a-x)} = \frac{n!}{a}\left(\frac{(-1)^n}{x^{n+1}}+\frac{1}{(a-x)^{n+1}}\right).\tag{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Math and geometry software to create instructional videos I am looking into starting an amateur project for online tutoring. I need a math/geometry program that I can use to create shapes, graph functions, and create animation. I know of many types of software (such as geogebra) but they have restrictions for commercia... | You can take a look at GNU Dr. Geo a free software of mine, there is no commercial limitation and you can even redistribute it along your work.
You can do classic interactive geometry with:
Or you can use its programming features to design very original sketches:
More over, it is very easy to modify Dr. Geo from itse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The quotient map and isomorphism of cohomology groups Let $X$ be a closed $n$-manifold, $B$ an open $n$-disc in $M$.
Suppose $p:X\rightarrow X/(X-B)$ is a quotient map. Notice that $X/(X-B)$ is homeomorphic to the sphere $\mathbb{S}^n$.
My question is whether the quotient map induces an isomorphism mapping $H^n(X;\ma... | Consider the map of pairs $(X,X-B)\to(S^n,*)$ with $S^n=X/(X-B)$ and $*$ the point resulting from collapsing $X-B$, and the diagram you get from it in the long exact sequences for cohomology. The map $H^n(S^n,*)\to H^n(S^n)$ is an isomorphism, and so is $H^n(S^n,*)\to H^n(X,X-B)$, so by the commutativity of one of the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I have a question about Viete's formulas If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as,
$a_n x^n + a_{n-1}x^{n-1} \cdots + a_1 x + a_0 = (x-r_1)\cdots(x-r_n)$. Now if I wanted to express each coeffi... | The answer to both questions is yes, provided that $a_n=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. Show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$.
My proof: let $(x_n)$ be Cauchy in $(X,e)$. Let $(t_n) := \frac{1}{(x_n)}$. Then $(t_n)$ is ... | The idea is OK, but should be formalised a bit more.
Let $Y = [1,\infty)$ with the Euclidean metric $d$. Clearly, $Y$ is complete
as it is a closed subset of the complete $\mathbb{R}$ in the Euclidean metric.
Then $f(x) = \frac{1}{x}$ from $(X,e)$ to $(Y,d)$.
Then $d(f(x), f(y)) = |f(x) - f(y)| = |\frac{1}{x} - \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Easy: Graphs of Straight Line
I can't exactly figure out how to work this out.
Well I know the equation for a straight line is $y = mx + c$
$c = gradient$
Therefore if I multiply $3$ by the number $x$ to get the gradient $6$ and $2$ I can work out which is line is which...
With $y = 3(x + 2)$ I tried to expand but the... | Syntactical side note: $m$ is called the gradient, $c$ is called the $y$-intercept.
To find the correct equation for $A$, just look at the $y$-intercept of $A$: it's $6$.
This means that $A$ must be of the form $y = mx + 6$, wich only one of your equations satisfies. Note that
$$a\cdot(b+c) = a\cdot b + a\cdot c$$
for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Subring of a field extension is a subfield For the first part,
I am trying to show that any subring of $E/F$ is a subfield where $E$ is an extension of field $F$ and all elements of $E$ are algebraic over $F$.
My solution is to just show an element of a subring has an inverse and we will be done. Let $R$ be the sub... | Let $R$ be a ring such that $F\subset R\subset E$ and $\alpha\neq 0$ an element of $R$. We can as well suppose $R=F[\alpha]$. Then multiplication by $\alpha$ is an endomorphism of the finite dimensional $F$-vector space $R$. This endomorphism is injective because $R$ is an integral domain (subring of a field), hence it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you helped me with finding the limit, hinted or thorough.
| $$
\lim_{x\uparrow1} \ln x \ln(1-x) = \lim_{x\uparrow1} \frac{\ln x}{x-1}\cdot\frac{\ln(1-x)}{1/(1-x)}.
$$
Apply L'Hopital's rule to both fractions and you've got it. (In the first one, the numerator and denominator both approach $0$; in the second, each approaches either $+\infty$ or $-\infty$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Recover the inverse after interative solution of a linear system I have solved the linear system $\mathbf{A} \mathbf{x} = \mathbf{b}$ with an iterative solver. The problem is well-posed ($\mathbf{A}$ is invertible, $\mathbf{b} \ne \mathbf{0}$, blah blah blah).
The question: Is it possible to combine $\mathbf{A}$, $\ma... | Each columns of the inverse is the solution to $Ax = b$ when $b = e_1,e_2,\dots,e_n$, where $e_i$ are the standard basis vectors.
At the very least, finding $A^{-1}$ from the $x$'s would require that you choose $n$ linearly independent $b$s. Perhaps you can optimize on which $b$'s you select.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convert 1-5 Grading Scale to 1-100 Grading System I am creating a formula in Excel to convert 1-5 Grading Scale to 1-100 Grading System Suppose that I have the following table:
97-100 = 1.00
94 - 96 = 1.25
91-93 = 1.50
88-90 = 1.75
85-87 = 2.00
82-84 = 2.25
79-81 = 2.50
76-78 = 2.75
75 = 3.00
My problem is this:... | This is an answer based more on intution. Hopefully, it will be easier to understand.
You can multiply the score in the ratio of their respective range.
So, since the ratio will be $\frac{100}{5}=20$.
Let $S_5$ and $S_{100}$ be the respective scores.
Therefore,
$$S_{100}=S_5 \times 20$$
However, I included $0$ here. Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluating $\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx$ How do I evaluate the definite integral $$\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx ?$$ I used trig substitution, and then a u substitution for $\sec\theta$.
I tried doing it and got an answer of: $-\sqrt{125}+12\sqrt{5}-16$, but apparently its wrong.
Can someone hel... | A way to compute this is as follows:
\begin{align*}
\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\mathrm d x &=\int_0^1\frac{4x+x^3-4x}{\sqrt{4+x^2}}\mathrm d x\\
&=\int_0^1x\sqrt{4+x^2}\mathrm d x -2\int_0^1\frac{2x}{\sqrt{4+x^2}}\mathrm d x\\
&=\left.\frac{1}{3}(4+x^2)^{3/2}\right|_0^1-4\left.\left(4+x^2\right)^{1/2}\right|_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Clarification of some doubts: working with the restriction of a quadratic form Let $q:\mathbb{R^3}\to\mathbb{R}$ such that
$$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$
I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and $\operatorname{sgn}(q)= (2,1)$. The second question ... | That plane $V$ is described by $z=0$, hence your restriction is just
$$q(x,y,0)=2x^2+3y^2+4xy$$
This is then
$$q'(x,y)=2x^2+3y^2-4xy$$
which is represented by the matrix
$$\begin{pmatrix} 2 & 2 \\ 2 & 3\end{pmatrix}$$
The eigenvalues are ${5\over 2}\pm {\sqrt{17}\over 2}$, which are both positive.
so the new signature ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$. first of all I checked if (0,0) is critical point
$Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $
now my idea was to replace $y$ with $xk$ because of the reduction of $F$
,and find the hessian ma... | A direct way to prove that $(0,0)$ is a local minimum for $F$ is just to notice that $F(0,0)=0$ and that
$$
\begin{align}
F(x,y)-F(0,0)&=2x^4-3x^2y+y^2\\\\
&=2\left( x^4-\frac32x^2y+y^2\right)\\\\
&=2\underbrace{\left( \left(x^2-\frac34y\right)^2+\frac7{16}y^2\right)}_{\large\geq \:0}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditions for a supremum of a set. Suppose a function $f(x)$ is continuous on $[a, b]$ and there exists, $x_0 \in (a, b)$ such that $f(x_0) > 0$. And then define a set,
$$A = \{ a \le x < x_0 \space | \space f(x) = 0 \}$$
We say $c = \sup A$
Does $c$ necessarily have to suffice that $f(c) = 0$ ? ? Why/ why not?
I a... | Yes, provided that $A$ is not empty: Since $c$ is the supremum of $A$, we find a sequence $(x_n)\subset A$ of elements of $A$ converging to $c$. Since each number $x_n$ is an element of $A$, we must have $\color{blue}{f(x_n)=0}$. Since $f$ is $\color{green}{\text{continuous}}$, we conclude
\begin{align*}
f(c)=f\left(\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1124924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Completing simplification step when solving a recurrence I am trying to understand a simplification step in one of the recurrence examples solved by repeated substitution in a book of algorithms problems I found on Github. I am using it for extra practice in solving recurrences. Anyways, here's the step I am stuck on:
... | $$3^{i-1}\color{green}(3T(n-i)+2\color{green}) + 2\sum_{j=0}^{i-2}3^i$$
$$\color{green}{3^{i-1}3}T(n-i)+3^{i-1}\color{green}2 + \color{green}2\sum_{j=0}^{i-2}3^i$$
$$3^iT(n-i) + 2(\color{green}{3^{i-1}+\sum_{j=0}^{i-2}3^i})$$
$$3^iT(n-i) + 2\sum_{j=0}^{i-1}3^i$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find all planes which are tangent to a surface I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes to the surface given one point, but how would I do so given two points? A... | Given $p_1,p_2$ and $G(p) = 0$ with $p = (x,y,z)$, determine a plane $\Pi\to (p-p_1)\cdot\vec n=0$ with $\{p_1,p_2\}\in \Pi$ and such that $\Pi$ is tangent to $G$ at $p^*$
Representing for $\nabla G(p^*)$ the normal to the the surface $G(p)=0$ at point $p^*$ we have the conditions.
$$
\cases{
\nabla G(p^*) = \lambda\ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
A relationship between central-by-finite groups and FC-groups A group is said FC-group if for all $x\in G$ is true that the set $x^G$ is finite. Equivalently, $G$ is a FC-group if $|G:C_G(x)|$ is finite for all $x \in G$.
A group is said a central-by-finite if the center of $G$ has finite index $G$.
Is clear that if $G... | A central product of countably infinitely many copies of $D_8$ (or more generally of any extraspecial group) is a counterexample. This group has the presentation
$$\langle x_i,y_i,z\ (i \in {\mathbb N}) \mid z^2=1,\ x_i^2=y_i^2=1,\ [x_i,y_i]=z\ (i \in {\mathbb N}),\
{\rm all\ other\ pairs\ of\ generators\ commute}\ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I prove that if $p$ is prime then $p$ divides $2^{p}-2$? I know that if $p$ divides $2^{p}-2$ can be written as $2^p - 2 \equiv 0 \bmod p$, but then I get stuck. Im not sure how to take an approach on this.
| $$(2\cdot1)\cdot(2\cdot 2)...(2\cdot (p-1))=1\cdot2 ...\cdot (p-1)\ \ \ \ (\text{ mod } p )$$
Where the factors on the left and right are equal but not necessarily in the order written.
The reason is that the remainders of $2\cdot i$ mod $p$ are all different (and different to zero) for different $1\leq i\leq p-1$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
What is the derivative of this? I have a function of the following form:
$J = \|W^TW-I\|_F^2$
Where, $W$ is a matrix and $F$ is the Frobenius Norm.
How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
| Assuming the norm comes from real inner-product. The derivative of the norm can be found as
$$\left.\frac{d}{dt}\right|_{0}\|(W+tH)^T(W+tH)-I\|^{2}\\
=\left.\frac{d}{dt}\right|_{0}\langle (W+tH)^T(W+tH)-I,(W+tH)^T(W+tH)-I\rangle\\
=2\langle W^TW-I,H^TW+W^TH\rangle$$
We now use the definition of adjoint corresponding to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Two questions about discrete valuation rings of varieties Let $X$ be a proper, normal variety over $\mathbb{C}$, and $k(X)$ be its field of rational functions. I think the following two statements are true, but I was unable to give a proof or find the references:
(1) For any $f \in k(X), f \neq 0$, there are only finit... | $\textbf{Wrong answer. See Comments!}$
$\def\OO{{\mathcal O}}\def\CC{{\mathbb C}}$I think the following argument shows that in fact $v$ is the valuation associated to a prime Weil divisor on $X$ from which part $(1)$ follows as well.
Let $v$ be such a discrete valuation. Consider the valuation ring $\mathcal{O}_v \sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Applying the Cauchy Schwarz Inequality Let $A = (a_{ij})$ be an $n \times n$ real matrix, $I = (\delta_{ij})$ the $n \times n$ identity matrix, $b \in \mathbb{R}^n$. Suppose that $$\|A - I\|_2 = \left(\sum_{i=1}^n \sum_{j=1}^n (a_{ij} - \delta_{ij})^2\right)^{\frac{1}{2}} < 1$$ 1) Use the Cauchy-Schwarz Inequality to s... | Write $Tx = b - (A - I)x$. Then you can see that $Tx - Ty = (A - I)(y - x)$, which implies $\rho(Tx,Ty) \le \alpha \rho(x,y)$, with $\alpha = \|A - I\|_2 < 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Alternative proof of a transpose property I am asked to prove; $$(AB)^T=B^TA^T$$ although it is very simple to prove it by the straight forward way, in the exercise I am asked to prove it without using subscripts and sums, directly from the following property of inner product of real vectors:$$\langle A\textbf{x},\text... | Use nondegeneracy of inner product. I.e if for every vector $x\in\mathbb{R}^n$
$$<x,y_1>=<x,y_2>$$
holds, then $y_1=y_2.$ Fix $y\in\mathbb{R^m}$ and from what you have quoted you have that for every $x$
$$<ABx,y>=<Bx,A^Ty>=<x,B^TA^Ty>$$
and
$$<ABx,y>=<x,(AB)^Ty>.$$
Hence $B^TA^Ty=(AB)^Ty.$ $y$ was arbitrary, hence $B^T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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