Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solving simultaneous equations with complex coefficients using real methods My circuits analysis textbook teases that there's a way to convert a set of n complex equations into a set of 2n real equations, which can then be solved using any calculator that can solve real simultaneous equations. That is, no capability wi... | The main idea is to split each equation into a real and a complex part.
To easily see how to do this take a look at complex multiplication as a linear transformation.
$(a + bi) * (c + di) = (ac - bd) + (bc + ad)i$ will become
$\left(\begin{array}{cc}c&-d\\d&c\end{array}\right)
\left(\begin{array}{c}a\\b\end{array}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Derivative function went wrong I am trying to take the derivative of this function but I am facing some difficulties.
$$f(x)= e^{\ln(e^{7x^2+11})}$$
My answer was : $7e^{(7(x^2))}*14x$
I cancelled the $\ln$ with the $e$ first, then I downgrade the $7$ and keep the $\exp$. as it is, after that I took the derivative of ... | $$f(x)= e^{\ln(e^{7x^2+11})}=\exp(\ln(\exp(7x^2+11)))$$
So
$$f(x)= \exp(7x^2+11), \implies f'(x)=14x\exp(7x^2+11)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/842311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $
Show that
$$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$
Indeed,
First let's show
$7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $
we've $7\mid x \implies 7\mid x^2$ the same for $7\mid y \implies 7\mid ... | $►$ If $x=7x_1$ and $y=7y_1$ then $x^2+y^2=7(x_1^2+y_1^2$).
$►$ If $x^2+y^2\equiv0\pmod7\iff x^2\equiv -y^2\pmod7$ then because of $\mathbb F_7^2=\{1,4,2,0\}$ and $(-1)\mathbb F_7^2=\{6,3,5,0\}$ the only possibility for
$x^2\equiv -y^2\pmod7$ is that both $x^2$ and $y^2$ are equal to $0$ modulo $7$ so $x$ and $y$ are e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
Soft sheaves adapted to $f_!$ I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to $f_!$, I have to show that every acyclic complex of soft sheaves is mapped by the functor $f_!$... | It is not needed in the proof, it just makes it easier. Because of that exercise you only need to prove that $f_!$ is right exact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/842473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that $$\lim\limits_{(h,k) \to (0,0)} \dfrac{|f(h,k)|}{|(h,k)|} = 0$$.
| For simplicity, I'll be using the euclidean norm for $\Bbb R^n$, $\Bbb R^m$ and $\Bbb R^n\times\Bbb R^m$, and the $\sup$ norm for the target space $\Bbb R^p$. The choice of norms is irrelevant to the result since all norms are equivalent in finite dimension.
There are coefficients $a_{ij;\,l}\in\Bbb R$, where $(i,j)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$
Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$
Prove:
*
*$\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 {\displaystyle\liminf_{n\to\infty}a_n}$
*$\displays... | To get the first one, you just need to notice that
$$\sup\{1/{a_k}; k\ge n\} = \frac1{\inf\{a_k; k\ge n\}}$$
and take limit for $n\to\infty$ to get
$$\lim_{n\to\infty}\sup\{1/a_k; k\ge n\} = \lim_{n\to\infty}\frac1{\inf\{a_k; k\ge n\}} = \frac1{\lim_{n\to\infty}\inf\{a_k; k\ge n\}}.$$
(Although you should also check th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
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Möbius transformation: proving the image of the unit circle is a line Problem
1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively.
2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$.
In $1)$ I didn't have problems, the homographic tr... | An element of $C$ is $e^{it}$ with $t\in\mathbb{R}$:
$$T(e^{it})=\frac{2e^{it}}{e^{it}+i}=\frac{2e^{it}(e^{-it}-i)}{2+2\sin(t)}=\frac{2-2ie^{it}}{2+2\sin(t)}=\frac{1-i(\cos(t)+i\sin(t))}{1+\sin(t)}=$$
$$=\frac{1+\sin(t)-i\cos(t)}{1+\sin(t)}=1-i\frac{\cos(t)}{1+\sin(t)}\ .$$
This is a parametrization of the line $Re(z)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why do some sources call calculus, "the calculus"? No need to cite specific sources since I think it's a fairly common thing to see. What's up with that?
Thank you
Edit: I've seen it in several places. Here's where I'm currently looking at it at:
Calculus: An Intuitive and Physical Approach (Second Edition): Here on ... | Because 'calculus' meant a set of rules for calculating things whereas 'the calculus' meant 'the infinitesimal calculus'. The qualifier was lost in the academic war over the foundations of the subject.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/842907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Understanding trig interval I have kind of a random question I'm hoping someone could help me with.
So I was thinking about the interval $[-\pi, \pi]$ for a trig functions. Isn't this is the same interval as $[0, 2\pi]?$ The reason why I say that (and maybe this is where my confusion is) is because couldn't $[-\pi, \pi... | I feel it would depend on the specific trig function only if you include hyperbolic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/843057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Related to the construction of $\Bbb C$ (generalisation) To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$
then write $(0,1_{\Bbb R}) = i$, go on writing $(a,b)$ as $a+ib$, etc. Maybe the questi... | Close. To get the complexes, use real matrices
$$
\left( \begin{array}{rr}
a & b \\
-b & a
\end{array}
\right) .
$$
To get the quaternions, use complex matrices
$$
\left( \begin{array}{rr}
\alpha & \beta \\
-\bar{\beta} & \bar{\alpha}
\end{array}
\right) .
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/843159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd? Let G be a finite group G. Then How can I show that no. of elements $x$ of group $G$ such that $x^3=e$ is odd ?
I read this question in an Algebra book. Since $e^3=e$, e must be one of those elements. But how to find for non trivial elements ?
| We claim that the number of elements in $T = \{g\in G: g^3 = e, g\neq e\}$ is an even number $N$. The claim follows from recalling that $e^3 = e$, hence the number of elements with trivial cubes is $N+1$, an odd number.
In fact, if $g\in T$, then $\{g,g^2,g^4...\}\subset T$, so, since $T$ is finite, there is a minimal ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Why is $(\sec x)' = \tan x\sec x$ and not $\tan x$? As far as I understood, the Fundamental Theorem of Calculus states that the integral of a function is its anti-derivative. And yet, although the integral of $\tan x$ is $\sec x$, the derivative of $\sec x$ is $\tan x\sec x$.
I understand the calculation and you get $... | The actual anti-derivative of $\tan{x}$ is:
$$\int\tan{x}\,\mathrm{d}x=\int\frac{\sin{x}}{\cos{x}}\,\mathrm{d}x=\int\frac{-\mathrm{d}(\cos{x})}{\cos{x}}=-\ln{(\cos{x})}+\text{constant}.$$
This gives us the definite integral,
$$\int_{0}^{x}\tan{u}\,\mathrm{d}u=-\ln{(\cos{x})}=\ln{\left(\frac{1}{\cos{x}}\right)}=\ln{(\se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two children paradox: where is my reasoning wrong? I hope here is the good place to be asking this. Apologies otherwise.
The statement is as follows:
"Ms Michu has two children. We know one of the two is a girl, we call that girl Ludivine. What is the probability that Ludivine has a brother, rather than a sister?"
This... | The probability of the situation you have described as "B(y) G(L)(e)" is $\frac{1}{4}$, as you have said. But the probability of "G(y) G(L)(e)" is $\frac{1}{8}$: probability that elder child is a girl, $\frac{1}{2}$; probability that younger child is a girl, $\frac{1}{2}$; probability that you give the name Ludivine t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
every continuous map $S^1 \rightarrow S^1$ can be extended to continuous map $B^2 \rightarrow B^2$ Let $S^1$ denote the unit circle, and $B^2$ denote the closed unit disk. I came across this question and got stuck:
Q:) Every continuous map $f :S^1 \rightarrow S^1$ can be extended to continuous map $B^2 \rightarrow B^2$... | Riccardo's comment is still useful you just need to be clever in how you apply it. Every continuous map $f\colon S^1\to S^1$ can be extended to a map $\tilde{f}\colon S^1\to B^2$ by composing with inclusion $i\colon S^1\to B^2$ of the boundary, so
$$\tilde{f}=f\circ i$$
and then because $B^2$ is contractible, $\tilde{f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Help! Totally stuck up with this limit.. I don't even know where to start with this
$$\displaystyle\lim_{x \rightarrow \frac{\pi}{6}} (2+\cos {6x})^{\ln |\sin {6x}|}$$
Please help me out (Hints in the right direction would be appreciated)
| When $f(x)\to1$ and $g(x)\to\infty$ then in order to find $\lim f(x)^{g(x)}$, first find $\log\lim f(x)^{g(x)} = \lim\log(f(x)^{g(x)})= \lim(g(x)\log f(x))=L$ and conclude that $\lim f(x)^{g(x)}=\exp L$.
In this case you I'd try L'Hopital's rule applied to $\dfrac{\log f(x)}{1/g(x)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/843590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Limit points of $\cos n$.
Find the limit point of the sequence $\{s_n\}$ given by $s_n=\cos n $.
I know by this post Limit of sequence $s_n = \cos(n)$ that the sequence does not converge. But I don't know how to search those points.
| The set $A=\{n+2\pi k:n,k\in\mathbb{Z}\}$ is dense on $\mathbb{R}$. Given a $y\in[-1,1]$ there existe an $x\in\mathbb{R}$ such that $\cos x=y$. Since $A$ is dense on $\mathbb{R}$ there existe a sequence $s_m=n_m+2\pi k_m$ of elements in $A$ such that $\lim\limits_{m\rightarrow\infty}s_m=x$. Then $$\lim\limits_{m\righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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Finite Group with Nilpotent Subgroup of Prime Power Index is Solvable Let $G$ be a finite group, and assume that $H$ is a nilpotent subgroup whose index is a prime power. WLOG, we can say that the index of $H$ is the highest power of $p$ which divides the order of $G$.
I want to show that $G$ is solvable, and I'm allo... | The main argument in the character theoretic proof of Burnside's $p^aq^b$ theorem actually proves that a finite simple group cannot have a conjugacy class of prime power order bigger than $1$. In your problem, $H$ is nilpotent, so it has nontrivial centre. Then, for $1 \ne h \in Z(H)$, either $h \in Z(G)$, in which cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist
Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist.
I think that if $f'$ isn... | The very fundamental thing one needs to observe here is that a derivative can't have jump discontinuity. If $f'(x) \to L$ as $x \to c$ then $f'(c) = L$ and thus $f'$ is continuous at $c$. Hence it is not possible for a derivative to have a limit at point and not to be continuous at that point. It follows that there wil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Is the set discrete?
Is the set $S=\{(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}):m,n,p,q\in \mathbb{Z}$} discrete in $\mathbb{R}^2$?
I'm not getting how shall I check discrete here?
| Hint: Take a particular point $P=\left(m+\frac{1}{2^p}, n+\frac{1}{2^q}\right)$. We want to show that the point is an isolated point. To do this, show that our set contains no point other than $P$ within distance $\min\left(\frac{1}{2^{p+2}},\frac{1}{2^{q+2}}\right)$ of $P$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/843871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Show that $\dfrac{\sqrt{8-4\sqrt3}}{\sqrt[3]{12\sqrt3-20}} =2^\frac{1}{6}$ This was the result of evaluating an integral by two different methods. The RHS was obtained by making a substitution, the LHS was obtained using trigonometric identity's and partial fractions.
Now I know that these two are equal, but I just can... | We have
$$8 - 4\sqrt{3} = 2(4-2\sqrt{3}) = 2(3-2\sqrt{3}+1) = 2(\sqrt{3}-1)^2,$$
so $\sqrt{8-4\sqrt{3}} = \sqrt{2}(\sqrt{3}-1)$.
Then we see that looking at $(\sqrt{3}-1)^3$ is a good idea:
$$(\sqrt{3}-1)^3 = (4-2\sqrt{3})(\sqrt{3}-1) = 6\sqrt{3} - 10,$$
so $12\sqrt{3}-20 = 2(\sqrt{3}-1)^3$ and $\sqrt[3]{12\sqrt{3}-20}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
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prove that there infinitely many primes of the form $8k-1$ Using the fact that $$\left ( \frac{2}{p} \right )=(-1)^{\frac{p^2-1}{8}}$$
for each prime $p>2$,prove that there infinitely many primes of the form $8k-1$.
I thought that we could I assume that there is a finite number of primes of the form $8k-1$: $p_1,p_2 \d... | Let $p_1,p_2, \ldots, p_k$ be the list of ALL primes of the form $8s+7$. Let
$$N=(p_1p_2 \dotsb p_k)^2-2.$$
Note that $N \equiv 7 \pmod{8}$ and is odd. If $p$ is a prime that divides $N$, then
$$(p_1p_2 \dotsb p_k)^2 \equiv 2 \pmod{p}.$$ Thus
$$\left(\frac{2}{p}\right)=1.$$
Thus $p \equiv \pm 1 \pmod{8}$.
So all prime... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Using continuity to evaluate limits I hope you guys are enjoying your weekend. I have a question about limits.
This homework problem asks me to use continuity to evaluate this limit, I would like to double-check that I have following the right procedure.
The problem is as follows:
$$\lim_{x\to \pi}\sin(x + \sin x)$$
I ... | Using continuity to evaluate the problem means that you can use the following fact (assuming you proved it in class, not sure what else your teacher might have been asking) for a continuous function $f:\mathbb{R} \to \mathbb{R}$:
$$\lim_{x\to a} f(x)=f(\lim_{x\to 0} x).$$
So $\lim_{x\to \pi}\sin(x+\sin x)=\sin(\lim_{x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Is this correct and sufficient to show limit does not exist? Find limit or show that it does not exist:
$$\lim_{(x,y) \to (0,0)} \frac{ 2x^{2}y^{3/2} }{y^{2}+x^{8}}$$
using the path $x=m y^{1/4}$:
$$\lim_{(my^{1/4},y) \to (0,0)} \frac{ 2m^{2}y^{1/2}y^{3/2}}{y^{2}+m^{8}y^{2}}$$ $$\lim_{(my^{1/4},y) \to (0,0)} \frac{ 2... |
Looks good to me! Good job. Intuitively you can guess that the limit won't exist because the overall power in the denominator dominates the power of the numerator. To realize this, you can look at straight-line paths to the origin by setting $y=ax$
and taking a limit that way. The numerator will have degree $\dfrac7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Basic Trigonometry Question If $\cos{(A-B)}=\frac{3}{5}$ and $\sin{(A+B)}=\frac{12}{13}$, then find $\cos{(2B)}$.
Correct answer = 63/65.
I tried all identities I know but I have no idea how to proceed.
| $$\cos(2B)=\cos(A+B+\underbrace{A-B})=\cos(A+B)\cos(A-B)-\sin(A+B)\sin(A-B)$$
$$A-B=\arccos\frac3{15}$$
Using the principal value, $$\implies 0<A-B<\frac\pi2\implies\sin(A-B)>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/844323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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True or False: The circumradius of a triangle is twice its inradius if and only if the triangle is equilateral. Let $R$ be the circumradius and $r$ be the inradius. The if part is clear to me. For an equilateral triangle, the circumcentre, the incentre and the centroid are the same point. So, by property of cebntroid $... | Let $a,b,c$ are the sides of a triangle, $A=$ area of the triangle, $s=$ semi-perimeter.
$R=\dfrac{abc}{4A}, r=\dfrac{A}{s}$
We have to show $R\geq 2r$.
The relation $\dfrac{abc}{4A}\geq \dfrac{2A}{s}$ holds
if $abc\geq \dfrac{8A^2}{s}$
if $abc\geq 8(s-a)(s-b)(s-c)$
if $abc\geq (b+c-a)(c+a-b)(a+b-c)$
This is true for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
2-Frobenius Groups of order 25920 A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with kernel $B$ and complement $C$.
Let $A$ be a nilpotent group of order $2^4.3^4=1296$ an... | The smallest dimensional modules for a cyclic group of order $5$ over both ${\mathbb F}_2$ and ${\mathbb F}_3$ have dimension $4$, so $A$ must be a direct product of elementary abelian groups of orders $2^4$ and $3^4$. Also, $BC$ has a single faithful $4$-dimensional module over both ${\mathbb F}_2$ and ${\mathbb F}_3$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Axiom of Pairing Axiom of Pairing states that if $a,b$ are sets, $\exists$ a set $A$ such that $A=\{{a,b\}}$. My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?
| User wrote:
My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?
To apply Specification to construct $A=\{a,b\}$, $a$ and $b$ would have to have been assumed or proven to be elements of some other set $B$, i.e. $a\in B$ and $b\in B$. Then $A=\{x|x\in B \land [x=a \vee x=b]\}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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how to solve this question of polynomials
Given the polynmial is exactly divided by $x+1$, when it is divided by
$3x-1$, the remainder is $4$. The polynomial leaves remainder $hx+k$ when divided by
$3x^2+2x-1$. Find $h$ and $k$.
This is the question which is confusing me.. i have done this question like this:
$p(... | The error is that $\,p(-1) = 0\,\Rightarrow\, h(-1) + k = \color{#c00}0,\,$ not $\,\color{#c00}4.\,$ Fixing that yields the given answer.
Remark $\ $ This is a special case of the Chinese Remainder Theorem (CRT) or, equivalently, Lagrange interpolation. Either of these methods can be applied to solve the general case.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The nonexistence of a polynomial I'm studying algebraic geometry. To illustrate a nonalgebraic set, it is given that a unit circle except for a point on it in cartesian product or whole plane except for one point. Why doesn't a polynomial whose zeros are a unit circle except for a point or whole plane except for one p... | Polynomials are continuous functions. A function that is identically zero on the plane except at one point is not continuous. The same goes for the unit circle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/844777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that mean KDR in a videogame is one This is not related to schoolwork. A friend of mine challenged me to prove that the mean KDR (assuming players can only die at the hands of other players) must always be equal to one. I have gotten through the logic part, and am now faced with the math part (which I am less c... | This is clearly false,if there are two player: Alice and Bob and Bob kills alice 20 times and Bob kills alice once then Alice's ratio will be $\frac{1}{20}$ and Bob's ratio will be $\frac{20}{1}$, the mean of these two ratio's is $\frac{401}{40}>20$
Construction to increase mean kdr arbitrarily for any number of playe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Textbooks, lecture notes, and articles from arXiv for undergraduate students I have found some interesting textbooks and articles on arXiv, such as the following one, that are accessible to an undergraduate student:
Course of linear algebra and multidimensional geometry, by
Ruslan Sharipov.
My experience makes me b... | I found this one a while ago:
Euclidean plane and its relatives;
a minimalist introduction
Anton Petrunin
https://arxiv.org/abs/1302.1630
btw...
Here is a "hack" you can do to look for books on arxiv. Do an advanced search for "these lecture notes" OR "this book" in the abstract. Show 200 results per page. Use your bro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Proof that $\lim_{n\to\infty} n\left(\frac{1}{2}\right)^n = 0$ Please show how to prove that $$\lim_{n\to\infty} n\left(\frac{1}{2}\right)^n = 0$$
| Notice that
$$
\frac{(n+1)/2^{n+1}}{n/2^n}=\frac12\left(1+\frac1n\right)\tag{1}
$$
For $n\ge2$, the ratio in $(1)$ is at most $\frac34$. At $n=2$, $\dfrac{n}{2^n}=\dfrac12$. Therefore, $(1)$ implies
$$
\frac{n}{2^n}\le\frac12\left(\frac34\right)^{n-2}\tag{2}
$$
for $n\ge2$. Hopefully, it is clearer that
$$
\lim_{n\to\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Question about a particular limit (at infinity) I have a question about limits, a problem specifically. I have been asked to solve the following limit in any way I see fit:
$$\lim_{x\to 2\pi^-}x\csc x$$
I know that the domain of $\csc$ is all numbers except for $n\pi$, and I know I could probably plug in numbers close ... | Hint: The limit is
$$\lim_{x \rightarrow {2\pi}^{-}}\frac{x}{\sin x}$$
For values of $x$ close to but less than $2\pi,$ the values will be
$$ \frac{\text{numbers close to} \;\; 2\pi}{\text{negative numbers close to} \;\; 0} $$
which tells us the limit is $\ldots$
(Note that $\sin x$ is negative when $x$ is in the $4$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$ There are two questions.
First: is the proof underneath correct?
Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$.
We will show that for any $\delta>0$ there exists $x$ and $y$ such that $|x-y|... | Your proof is fine, and the same proof works for any $f$ with $f' \to \infty$ as $x \to \infty$. Personally, I think your approach is excellent in that it is extremely clear. Perhaps you could have been more terse, but I like your style.
Another approach, however, if you insist on being as terse as possible, is to no... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Solve the integeral equation (C.S.I.R) Let $\lambda_1, \lambda_2$ be the eigen value and $f_1 , f_2$ be the coressponding eigen functions for the homogeneous integeral equation
$$ \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) dt = 0 $$
Then
*
*$\lambda_1 = -18 - 6 \sqrt{10} , \lambda_2 = -18 + 6 \sqrt{10}$
*$\lamb... | Let's rewrite your equation in the following form:
\begin{align}
\frac{1}{\lambda}\phi(x) = x\int_{0}^{1}t\phi(t) \mathrm{d}t + 2x^{2}\int_{0}^{1} \phi(t) \mathrm{d}t = c_{1}x + c_{2}x^{2}
\end{align}
for $c_{1}: = \int_{0}^{1}t\phi(t) \, \mathrm{d}t$ and $c_{2}:= 2\int_{0}\phi(t)\, \mathrm{d}t$. From this we can see t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Finding $\sin^{-1}(x)$ without using a calculator I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the necessary steps to solve the problem without a calculator.
Am I wrong to ass... | There are several approaches to this problem, all of them are a pain in the neck without a calculator. The first approach you can try is to construct a triangle which could give you a decent triangle or if you know some calculus, you can use the Taylor series of $\arcsin x$ and use that. I'll illustrate the Taylor seri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to establish this inequality without using induction? Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ for $n = 1, 2, 3, \ldots$?
I know how to establish the above ineq... | The fibbonacci numbers have a closed form: $a_n = \dfrac{1}{\sqrt{5}}\left[\left(\dfrac{1+\sqrt{5}}{2}\right)^{n+1} - \left(\dfrac{1-\sqrt{5}}{2}\right)^{n+1}\right]$.
Since $\left|\dfrac{1-\sqrt{5}}{2}\right| < 1$, we have $-1 < \left(\dfrac{1-\sqrt{5}}{2}\right)^{n+1} < 1$ for all $n \ge 1$.
Can you figure out what... | {
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Can $R[[x]]$ contain constants?
Consider the ring $R[[x]]$ of formal power series $\sum_{n=0}^\infty a_nx^n$ with coefficients in $R$. I was wondering whether $R[[x]]$ contains elements of $R$ (polynomials of degree $0$).
I'm trying to solve Commutative Algebra problems. I feel it is possible, as all of $\{a_1,a_2,\... | Yes, it can. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
| {
"language": "en",
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Criticism on truth of Gödel sentence in standard interpretation Mendelson in his book mentioned the Gödel sentence and argued that in standard interpretation it is true. But Peter Milne in his article (On Godel Sentences and What They Say) criticized: "But we know that we cannot move from consistency of T to truth of γ... | There is no real conflict.
Mendelson is talking about a particular type of Gödel sentence, constructed in a standard way following the pattern of Gödel's original paper. [This is the kind of sentence people usually have in mind when they speak of a Gödel sentence for a theory $T$, without further qualification -- the... | {
"language": "en",
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Equation , powers of two I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0 $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation which I couldn't solve. I think its roots sum to zero so I want to prove it without actually find... | $$4\left(a^2+\frac1{a^2}\right)-23\left(a+\frac1a\right)+40=0$$
$$\implies 4\left[\left(a+\frac1a\right)^2-2\right]-23\left(a+\frac1a\right)+40=0$$
$\displaystyle a+\frac1a=b\implies 4(b^2-2)-23b+40=0\iff 4b^2-23b+32=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/845674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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$\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always bounded Given the matrix $A= (a_{i,j}) \in M_{n,n}$
$||A||=\sup\limits_{x\in X}\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ where $|| $ . $|| _n$ is $ R^N$ norm
$\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always bounded:
$||Ax||^2_{n}=(\sum\limits_{i,j=0}^n a_{i,j} x_j)^2\le\sum\li... | We have
$$
\left\{ \frac{\|\mathcal Ax\|_{n}}{\|x\|_n} : x \in \mathbb R^n \right\}
=
\left\{ \|\mathcal Ax\|_{n} : x \in \mathbb R^n, \|x\|_n=1\right\}
$$
The set on the right is compact because $\mathcal A$ and norm are continuous and the unit sphere is compact. Hence, the set on the left is bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/845771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Normal subgroup Suppose $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$ that $|N|,[G:H]$ are finite and $(|N| , [G : H] ) =1$. Prove that $H\leq N$.
| Hint: $$|HN|=\frac{|H|\cdot|N|}{|H \cap N|} $$ and $[HN:N]$ divides $[G:N]$. But $[HN:N]=[H:H \cap N]$, which in its turn divides $|H|$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why there are irrational numbers? I do not quite get it. Why can't we represent all real numbers as a sum of rational numbers? Why do we need irrational numbers?
For example,
*
*$\pi=3.14159265358\cdots=3+10^{-1}+4*10^{-2}+10^{-3}+5*10^{-4}+\cdots$
*$e=2.71828182846\cdots=2+7*10^{-1}+10^{-2}+8*10^{-3}+2*10^{-4}+\cd... | Numbers that cannot be expressed rationally arise naturally as solutions of equations. The solution of $x^2=2$ & of $x(1-x)=1$, & any of an infinitude of specificable polynomial equations, cannot be expressed rationally. It can be proven directly, by a simple recipe, that no rational number can satisfy either of the tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
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World Cup probability question. What is the probability of none of the 32 teams of the World Cup bringing two consecutive draws in the first two games. Accept the fact that a win, draw or loss have the same probability to appear.
| Assume each game has a probability $p$ of ending in a draw, and that the results of each game are independent of each other. Lets analyze one group with teams W,X,Y,Z.
WLOG, the schedule for the first two games for each team is:
*
*W vs X
*Y vs Z
*W vs Y
*X vs Z
with the last two group games being W vs Z and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Valuation associated to a non-zero prime ideal of the ring of integers I have a question from Frohlich & Taylor's book 'Algebraic Number Theory', p.64. I will keep the notation used there.
Let $K$ be a number field, $\mathcal o$ its ring of integers. Let $\mathfrak p$ be a non-zero prime ideal of $\mathcal o$ and $v=v... | $v(x)$ is the number $n$ such that $x \in \mathfrak{p}^n-\mathfrak{p}^{n+1}$ now $v(x)\geq 1$ iff $x \in \mathfrak{p}$.
So we see that if $v(x^{\rho})=v(x)$ then
$$x^{\rho} \in \mathfrak{p} \Leftrightarrow x \in \mathfrak{p}$$
which means that $\mathfrak{p}^{\rho}=\mathfrak{p}$.
Conversely if $\mathfrak{p}^{\rho}=\mat... | {
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Show that some endomorphsm is not diagonalizable Given an endomorphism $f:V \rightarrow V$ on an $\mathbb{R}$-vector space, prove that if there is $v \in V-\{0\}$ such that $f^2(v)=-v$, then $f$ is not diagonalizable.
| Solved. I'm putting the solution. If A is a matrix for f in some basis, then $A^2v=-v$. If $A$ is in diagonal form, $A^2$ has only non-negative entries on the diagonal.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is $\mathbb{F}_4$ generated? I know $\mathbb{F}_4$ is a field while $\mathbb{Z}/(4)$ is just a ring. So how is $\mathbb{F}_4$ generated?
Complement: So what are the elements like in $\mathbb{F}_4$ like? Are they $\{0,1,x,x+1\}$? Is every field $\mathbb{F}_k$ has $k$ elements no matter whether $k$ is a prime?
| The answer of @MarceloBielsa is perfect, but I like an approach like what OP was working towards: the elements of $\mathbb F_4$ are $\{0,1,\omega,\omega+1\}$, where $\omega^2=\omega+1$, $(\omega+1)^2=\omega$, and $\omega(\omega+1)=1$. I think that that’s enough for filling in the multiplication table.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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help: isosceles triangle circumscribing a circle of radius r Please help me show that: the equilateral triangle of altitute $3r$ is the isosceles triangle of least area circumscribing a circle of radius $r$.
Iassumed the following:
base = $2a$
height = $h$
radius of circle = $r$
Area = $\frac{1}{2}(2a)h = ah$
$tan(2\th... | First, two corrections in the question
1) The height will be $3r/2$
2) This will maximize the area. Since the minimum area of an isosceles triangle will be of height $2r$ and zero base, hence zero area.
From the figure, $h=r+\sqrt{r^2-a^2}$, area $A=ah=a(r+\sqrt{r^2-a^2})$. Now equate $\frac{\partial A}{\partial a}=0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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1-form on Riemann Surface Good evening, I can not prove the following result:
Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show that f is ratio of polynomial functions.
Any suggestions on how to develop the demonstration? ... | As suggest in the comment, on another cover $\mathbb C$ (which corresponds to $\mathbb C\setminus\{0\} \cup \{\infty\}$ with coordinates $\tilde z$), write
$$\omega = \tilde f(\tilde z)$$
On the intersection of two coordinate $\mathbb C\setminus \{0\}$, we have $z = 1/\tilde z$, then
$$d\tilde z = d(1/ z) = -\frac{1}... | {
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Calculation of the limit $\lim_{n \to +\infty} n^2x(1-x)^n, x \in [0,1]$ and the supremum How can I find this limit:
$$\lim_{n \to +\infty} n^2x(1-x)^n, x \in [0,1]$$
Do I have to use the L'Hospital rule? If so, do I have to differentiate with respect to $n$ or to $x$ ?
EDIT:
I also tried to find the supremum of $n^2x(... | This limit is always zero, there are two cases:
1)$x=0$-it's obvious.
2)$x \neq 0$, then the limit is zero, because $1-x <1$ and exponential function decrease faster than polynomial $n^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/847019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Ways to sum to $n$ with $m$ integers that are $\le k$ Given three natural numbers $n$, $m$ and $k$, how many ways are there to write $n$ as the sum of $m$ natural numbers in the set $\{0, 1, \ldots, k\}$, where order does matter?
I've seen the "Ways to sum $n$ with $k$ numbers", but never where the different numbers ar... | It is not exactly what you want, but it is at least related:
OEIS A048887: Array T read by antidiagonals, where T(m,n) = number of compositions of n into parts <= m
What is missing is the condition that only those compositions are counted, which consist of exactly $m$ numbers, the above counts all.
So it could at least... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a \gg b > 0$. I know that, for $|x|\leq 1$, $e^x$ can be bounded as follows:
\begin{equation*}
1+x \leq e^x \leq 1+x+x^2
\end{equation*}
Likewise, I want some meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a \gg b > 0$.
The first thing that comes to my mind is $\sqrt{a... | The first three terms of $(1+x)^{\frac12}$ are $1 + \frac12 x - \frac18 x^2$. And you can check for yourself that $$\left(1 + \frac12 x - \frac18 x^2\right)^2 = 1 + x - \frac18 x^3 + \frac{1}{64}x^4$$
which is $\le 1+x$ whenever $\frac18 x^3 \ge \frac{1}{64}x^4$, i.e. for $0 \le x \le 8$.
Now just put $x = \frac{b}{a^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $4k^3+6k^2+3k+l+1$ and $4l^3+6l^2+3l+k+1$ are powers of two, how to conclude $k=1, l=2$ It is given that
$$4k^3+6k^2+3k+l+1=2^m$$
and
$$4l^3+6l^2+3l+k+1=2^n$$
where $k,l$ are integers such that $1\leq k\leq l$.
How do we conclude that the only solution is $k=1$, $l=2$?
I tried subtracting the two equations to get:... | I have a simplification that I couldn't finish, but maybe someone better with polynomials can. It may also lead nowhere, obviously; but was too long as a comment.
If you add the two equations, you get
$$4(k^3 + l^3)+ 6(k^2 + l^2) + 4 (k + l) + 2 = 2^m + 2^n$$
If you look at this modulo (k+l), it implies (mod k+l):
... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the domain of the function $F(x)=\int_{0}^{x}\frac{\operatorname{arctan}(t)}{t}dt$? What is the domain of the following function?
$$F(x)=\int_{0}^{x}\frac{\operatorname{arctan}(t)}{t}dt$$
On the one hand, the internal function is not defined at 0, but on the other hand, it's defined on every other point except ... | The function can be extended in $x=0$ to obtain a continuous function defined on the whole real line. Hence such extended function is Riemann integrable and hence its integral on $[0,x]$ is well defined for all $x$. Now remember that if you modify a function in a finite number of points, its integral does not change. H... | {
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Permutation, Combinatorics Stuck here : there are 100 objects labeled 1, 2,...100. They are arranged in all possible ways. How many arrangements are there in which object 28 comes before object 29.
My approach : Consider object 28 & object 29 , a single object. Now we have a total of 99 objects which can be per mutated... | You can take object 28 and 29 as alike. So the permutation order i.e. 28 before 29 is not disturbed. This gives no. of arrangements as 100!/2! = 98!*99*100/2=98!*4950
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Intuition Behind Compactification I'm heading into my second semester of analysis, and I still don't have a good intuition of when a set is compact. I know two definitions, covering compact and sequentially compact, but both of those seem very difficult to apply "real time". In $\mathbb{R}^n$ I know we have Heine-Borel... | There's a few different questions going on here, but I'll focus on the last one:
Yes, it is possible to compactify any space. An easy way to do so is to take your space $X$ and add a point called "$\infty$", and we say that a set $G$ containing $\infty$ is open if and only if $(X \cup \{\infty\}) \setminus G$ is a com... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The limits of the sum of functions whose limits do not exist I have a homework problem that I'm not sure how to start. I tried Google for similar examples it didn't turn up anything. Could someone tell me the name of the concept to look into? The problem is as follows:
Show by example that $\lim_{x\to c}f(x) + g(x)$ ca... | What about $f(x)=x$ and $g(x)=-x$. Neither limit as $x\to\infty$ exists as a real number, yet the limit of the sum is $0$. Or $f(x)=sinx$ and $g(x)=-sinx$. This time neither limit exists (as a real number OR +/-$\infty$, but the limit of the sum is $0$.
To get this phenomenon as $x$ approaches some real number $c$, ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sequences where each number is a divisor of one less than the next Let $N,k$ be fixed. Call a sequence of positive integers $a_1,a_2,\dots,a_k$ good if for each $i$, $a_i$ is a divisor of $a_{i-1}-1$. Consider the set
$$S = \{x : \text{$x$ occurs in some good sequence of length $k$ that ends in $N$}\}$$
of numbers th... | I'm sure sharper things can be said, but here are some estimates to calibrate thinking.
Let $f_k(N)$ be the function you describe. Note that $f_1$ is identically $1$, while $$f_k(N) = \sum_{d\mid(N-1)} f_{k-1}(d)$$ for all $k\ge2$. So for example, $f_2(N) = \tau(N-1)$ where $\tau$ is the number-of-divisors function.
Al... | {
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Drawing previously undrawn cards from a deck Suppose you have a deck of $y$ cards. First, randomly select $y-x$ distinct cards and sign the face of each, then shuffle all the cards back in to the deck. Proceed as follows:
Draw a card. If it is already signed, replace the card and shuffle the deck. If it is not yet sign... | From the description of the problem, we can set up the following recurrence:
\begin{align*}
p(n,x) &= \left(1-\frac{x}{y}\right)p(n-1,x)+\frac{x}{y}q(n-1,x-1) \\
q(n,x) &= \left(1-\frac{x}{y}\right)p(n-1,x)+\frac{x}{y}q(n-1,x-1) \\
p(0,x) &= 0 \\
q(0,x) &= 1 \\
p(n,0) &= q(n,0) = 0
\end{align*}
and the requir... | {
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"source": "stackexchange",
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What is the significance of the slope of the tangent line of a function? Why is the derivative so important? As I finished calc 1. I can use the product rule and chain rule and resolve integrals. But I feel like its too mechanical for my taste. I know the procedure and I execute on paper without really understanding or... | The differential is the slope or the rate of change.
On a roller-coaster:
*
*Your velocity is the rate of change of position.
*Your acceleration is the rate of change of your velocity.
*Your jerk is the rate of change of your acceleration.
*Your jounce is the rate of change of your jerk.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/847838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 6
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Lp space and sequence For what value of $p$ the sequence $\displaystyle x_{n}=\frac{1}{n}$ is on $l^p$ (where $\displaystyle l^p = \lbrace (x_1,x_2,...)| x_{i}\in\mathbb{C}\hspace{0.1cm}\text{and}\hspace{0.1cm} \left(\sum_{i=1}^{\infty}|x_{i}|^p\right)^{1/p}<\infty\rbrace$).
| If p=1, the series diverge. If p>2 it converges, so $x_n \in l^p$ for all $p\ge 2$ (since $l^p \subset l^{p+1}$)
| {
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Can zero divisors be in the denominator when we localize rings? Can we localize rings with zero divisors? Can those zero divisors be in the denominator?
I thought defining $$\frac{a}{b}=\frac{c}{d} \text{ iff }t(ad-bc)=0 \text{ where $b,d,t$ belong to the same multiplicative system}$$
accommodated for that little detai... | Your definition is correct and even necessary: if you don't include the factor $t$ into the definition, then you will in general not get an equivalence relation between pairs (a,b) of ring elements. This however is necessary to define the notion of a fraction a/b.
| {
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Derivative of a function is the equation of the tangent line? So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the ... |
So what exactly is a derivative?
The derivative is instantaneous (i.e. at any given precise moment in time) the rate of change of a dependent variable (usually $y$) with respect to the independent variable (usually $x$).
For straight lines, the derivative is simply the slope or gradient of the line.
For curves, whose... | {
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Parametric form of curve $\vert z+i\vert = 1$ I need to integrate a complex function through the curve $\vert z+i\vert = 1$.
As far as I know I need the parametric form of this curve.
I know that when I have $\vert z\vert = 1$, the parametric form is something like $\cos(t) + i\sin(t)$.
But what's different when I have... | $$
|z-z_0| = r
$$
is the equation of a circle centered in $z_0$ with radius r. Its parametric form is
$$
z = z_0 + re^{it} = z_0 + r(\cos t + i \sin t)
$$
| {
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How were 'old-school' mathematics graphics created? I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today:
Does anybody know how this graphic was created? Were equations used for... | Often the illustrations were drawn by hand, by the mathematicians themselves. The book A Topological Picturebook by George K. Francis (Springer, 1987) describes how one learns to do this:
This book is about how to draw mathematical pictures. … Theirs [the geometers of the 19th century] was a wonderfully straightforwa... | {
"language": "en",
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"question_score": "119",
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How to derive the formula to calculate the amount of cubes in a pyramid? The pyramid looks like:
For which I managed to derive the formula for the count of cube sides (ignoring the top). This was easy by simply thinking about it as a triangle:
If we have 4 squares wide pyramid, then the total sides represented graphic... | Consider each level separately. On the $k$-th level ($k$ starting at $1$ and counting from the top of the pyramid), there are $k^2$ blocks. Hence we just have to compute $1^2+2^2+\cdots+n^2$.
There is a standard result that
$$\sum\limits_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}=\frac{2n^3+3n^2+n}{6},$$
which is provable by in... | {
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How to tell if a Fibonacci number has an even or odd index Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$?
I know you can use the Pisano period, however if $n \equiv 1$ or $n \equiv 2$ $\mod \pi(k)$, it can never be found, where $k$ is... | Assuming that $F_n\geq 2$, you can check the parity of $n$ depending on the sign of the difference between $\frac{1+\sqrt{5}}{2}F_n$ and the closest integer. If negative, then $n$ is even, if positive, then $n$ is odd.
| {
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"source": "stackexchange",
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Compact $\omega$-limit set $\Rightarrow$ connected Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is compact, then it is connected? I think one should assume it is connected a... | So... After some more thinking I got a proof.
Assume $L_{\omega}(x)$ is not connected. So there are disjoint open sets $A,B\in \mathbb{R}^n$ such that $L_{\omega} \subset A \cup B$ and $A \cap L_{\omega},B\cap L_{\omega}$ are non empty. Therefore, there are sequences $\{t_n\},\{s_n\}$ such that $\displaystyle\lim_{n \... | {
"language": "en",
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"source": "stackexchange",
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Algebraically, What Does $\Bbb R$ get us? In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$?
Say we start with $\Bbb N$: $\Bbb N$ is closed under addition and multiplication. But then we decide we'd like... | I thinks the motivation for $\mathbb{R}$ is not algebraic, but rather it corresponds to our geometric intuition about which numbers are possible. Also it was founded at a time when we viewed the universe as continuous space.
| {
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"source": "stackexchange",
"question_score": "9",
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Minimize Sum a_i / Sum b_i over subsets I have two positive finite sequences $a_i$ and $b_i$, with $0 \leqslant i \leqslant n$. The problem is to find the subset $I$ of $\{0, ..., n\}$ that minimizes:
$$\frac{\sum_{i \in I} a_i}{\sum_{i \in I} b_i}$$
in an efficient way from the algorithmic point of view. Have you any ... | Hint: suppose you have some subset $I$ with a ratio $\frac ab$ and add one more element to make a subset $I'$. Does the ratio increase or decrease?
| {
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"source": "stackexchange",
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Ways to study mathematics while commuting I spend approximately 3 to 4 hours on public transport everyday. I try to maximize the usage of this time by checking email etc on my phone.
Are there any tips to study mathematics while commuting?
Thanks for sharing! Just want to make full use of the time spent commuting!
| Go to Amazon and search for whatever branch of mathematics you're looking for. Then change the order of the books to "average customer review" and you'll have a massively curated list of the best books in that topic.
| {
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Integration of exponential and square root function I need to solve this $$\int_{-\pi}^{\pi} \frac{e^{ixn}}{\sqrt{x^2+a^2}}\,dx,$$
where $i^2=-1$ and $a$ is a constant.
| By definition, $\displaystyle\int_0^\infty\frac{\cos x}{\sqrt{x^2+a^2}}dx=K_0\big(|a|\big)$, where K is a Bessel function. Letting $x=nt$, we
have $\displaystyle\int_0^\infty\frac{\cos(nt)}{\sqrt{t^2+a^2}}dt=K_0\big(|an|\big)$. Unfortunately, there are no “incomplete” Bessel functions,
so your integral does not posse... | {
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Distribution related to brownian bridge Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue when $t$ is replaced by random variable $U$. Could anyone help me?
| Let $p_t$ denote the PDF of $B(t)$ and assume that $U$ is independent of $B$ with PDF $f_U$, then the distribution of $B(U)$ has PDF
$$
q(\ )=\int p_t(\ )f_U(t)\mathrm dt.
$$
In the present case, $U$ is uniform on $(0,1)$ and, for every $t$ in $(0,1)$,
$$
p_t(x)=\frac1{\sqrt{2\pi t(1-t)}}\mathrm e^{-x^2/(2t(1-t))},
$$... | {
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Diameter of a circle using 3 nonlinear points I am trying to find the diameter of a circle using 3 points on its circumference. 2 of the points are 5 feet from eachother while the third point is centered between the other 2. The ceter point is 1 foot from a line drawn between the other 2 points.
| Hint: If two chords $AB$ and $CD$ of a circle intersect at $P$, then $AP\cdot PB=CP\cdot PD$. Draw the diameter joining the red and green lines in your diagram. So, $AP=PB=\dfrac{5}{2}$ and $CP=1$, and hence $DP=\dfrac{25}{4}$. Now you can compute the diameter.
| {
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Finding the basis of an intersection of subspaces
We have subspaces in $\mathbb R^4: $
$w_1= \operatorname{sp} \left\{
\begin{pmatrix} 1\\ 1 \\ 0 \\1 \end{pmatrix} ,
\begin{pmatrix} 1\\ 0 \\ 2 \\0 \end{pmatrix},
\begin{pmatrix} 0\\ 2 \\ 1 \\1 \end{pmatrix} \right\}$,
$w_2= \operatorname{sp} \left\{
\begin{pmat... | Hint: the intersection of these two spans is NOT empty. What you need to do is find a new spanning set for $w_2$ that contains some of the vectors from the spanning set for $w_1$. The common vectors will span the intersection. Now that you have a basis for $w_1\cap w_2$, you can extend it to a basis of $w_1+w_2$ by add... | {
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Periodic continuous function which is integrable on $\mathbb{R}$ Let $f:\mathbb{R}\to\mathbb{R}$ be a $T$-periodic function, that is $f(t+T)=f(t)$ for all $t\in \mathbb{R}$. Assume that
$$\int_0^{+\infty}|f(s)|ds<+\infty.$$
Now if we assume in addition that $f$ is continuous, my intuition tells me that we must have ... | This is correct. The way you can see this is by considering the maximum of $|f|$, call it $L$. For any $x$ such that $|f(x)|=L$, we have that $|f(y)| > \frac{L}{2}$ for all $|x-y| < \delta$ (for a sufficient choice of $\delta$). Can you see how to argue it from here?
| {
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Areas of contemporary Mathematical Physics I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc. have had a significant impact on pure mathematics especially geometry and conversely.
I am interested in knowing a list of areas of Mathematical... | Here is the 2010 Mathematics Subject Classification list. It has 6500 entries, working out the mathematical physics projection operator is left as an exercise to the reader.
| {
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Lagrange Bürmann Inversion Series Example I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for examples of this, but all I have found is the wikipedia article for the topic ... | An example of solution of a transcendental equation by means of Lagrange inversion can be the following.
Consider the transcendental equation:
$$(x-a)(x-b) = l e^ x $$
You can rewrite as:
$$x = a+ \frac{l e^ x}{(x-b)} $$
Applying Lagrange inversion:
$$x = a+ \sum_{n=1}\frac{l^n}{n!}\left[\left(\frac{d}{dx}\right)^{n-1... | {
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Area of a spherical cap formed by the plane containing any side of an inscribed regular tetrahedron I was trying to think about this problem today and realized that practically all of my high school geometry has deserted me, so "how to find it" answers would be greatly appreciated. As to the actual problem: Imagine tha... | (Another hint) Find the distance from the center of the inscribed tetrahedron to the center of one of its faces, call that $r$ [Once one can get the coordinates of some regular tetrahedron's vertices and center, after rescaling one can get this $r$.]
Once that $r$ is known, there is likely an available formula on-line ... | {
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Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function. This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of
$$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$
and shows that it is always $\leq 0$. Ho... | Differentiating is a clumsy way of solving the problem. However, let's look at the derivative. It is equal to
$$-\frac{x(0.5x+1)}{(x^2+x+1)^2}.$$
The denominator is bounded away from $0$. The numerator is negative for $x\gt 0$. Thus (Mean Value Theorem) our function is strictly decreasing in the interval $(0,\infty)$,... | {
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What am I supposed to do here now? $\tan(\pi/8) = \sqrt{2} -1$ complex analysis
Find $\sqrt{1+i}$, and hence show $\tan(\pi/8) = \sqrt{2}-1$
Okay so I know that $\sqrt{1+i} = 2^{1/4}e^{i\pi/8}$ and I know $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix} + e^{-ix}}{2}$
If i directly substitute those d... | You can work it out by rationalizing *
You can use a half-angle formula for Tan, i.e., a formula for Tan(B/2)
$tan(B/2) = (1 − cos B) / sin B = sin B / (1 + cos B)$
For CosB=SinB =$\sqrt \frac{2}{2}$, then , $Tan \pi/8= \frac{\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}=\frac{\sqrt2}{2+\sqrt2}=\frac{2\sqrt2-2}{2}=\sqr... | {
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Induced Lie algebra homomorphism from Lie group homomorphism: left translation A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$.
Which Lie algebra homomorphism is induced by left (or right)-translations:... | Left and right translations are not Lie group homomorphisms; they don't even preserve the identity, and the induced map on Lie algebras is obtained by looking at derivatives at the identity. However, conjugation by a fixed element $g \in G$ is, and the induced map on $\mathfrak{g}$ gives a representation $G \to \text{A... | {
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Group homomorphism $f$ is surjective iff $g$ is Let $G$ be an additive group, and let $u, v:G\to G$ to be two endomorphisms. Define $f(x) = x- v(u(x))$ and $g(x) = x-u(v(x))$. The question is to show that $f$ is surjective iff $g$ is.
I'm only able to show that $u:\ker f\cong \ker g$, but unable to show the statement ... | We have the following general fact: In a ring $R$ (not assumed to be commutative) with elements $u,v$, if $1-uv$ is invertible, then $1-vu$ is invertible. In fact, one checks that
$(1-vu)^{-1} = 1 + v (1-uv)^{-1} u$
(This formula has a nice motivation using the geometric series.)
We may apply this to the ring $R=\math... | {
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Show that this matrix is invertible I have the following exercise:
Show that the matrix $A=(a_{ij})$ where $a_{ij}=i^{j-1}$, $i,j=1, \dots n$ is invertible.
Do I have to show that the determinant is equal to $0$?
| From the wiki article on Vandermonde matrices, the determinant of the mentioned matrix would be
$$det = \prod_{1 \le i < j \le n}(j - i)$$
But since $j > i$, the above product will always be positive, i.e. not equal to $0$.
| {
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Please check my solution of $\int \sin^6(x)\cos^3(x) dx$ $$\int \sin^6(x)\cos^3(x) dx = \int \sin^6(x)(1-\sin^2(x))\cos(x)dx$$
$$\int \sin^6(x)\cos(x)dx - \int\sin^8x\cos xdx$$
Now, $\cos xdx = d(\sin x)$
$$\int u^6du - \int u^8du = \frac{1}{7}u^7 - \frac{1}{9}u^9 + C$$
$$\frac{1}{7}\sin^7(x) - \frac{1}{9}\sin^9(x) + C... | $11 + 7 \cos 2x = 11 + 7 - 14\sin^2 x = 2(9 - 7\cos^2 x) \Rightarrow\\
\dfrac {1}{126} (11 + 7 \cos 2x) = \dfrac{1}{63}(9 - 7\sin^2x) = \boxed{\dfrac{1}{7} - \dfrac{\sin^2x}{9}}\Rightarrow\\
\\
\text{The expressions are equal.}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/850618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The Cesàro Mean Theorem in the infinite case. I am trying to prove the Cesàro Mean Theorem in the infinite case. Let me state my problem more precisely.
Problem. Let $ (a_{n})_{n \in \mathbb{N}} $ be a sequence in $ \mathbb{R} $ such that $ \displaystyle \lim_{n \to \infty} a_{n} = \infty $. Then prove that
$$
\lim_... | Since $a_{n}\to\infty$ as $n\to\infty$ then there exists $N$ such that if $n\ge N$ then $a_{n}\ge M>0$. This means at most finitely many terms are negative. By choosing $N$ large enough we may also assume that $\sum_{k=1}^{n}a_{k}>0$ for $n\ge N$. By perhaps choosing $n$ even larger we may assume that $\lvert\frac{\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
how to derive the fact that the integral of $1/\sin^2(x) = -\cot (x)$ I know how that the integral of $\dfrac{1}{\sin^2(x)} = -\cot (x)$, but how does derive this fact? Can you use half-angle formula to do this integral?
| When you asked how to "derive" the fact that $\int \frac{1}{\sin^2 x} dx = -\cot x$, I thought maybe someone had suggested that this fact might be true, and asked if you could prove it. If that were so, then you would merely need to differentiate $-\cot x$.
In fact I would not be at all surprised to learn that the firs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Matlab Code to simulate trajectories of Ito process. I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated..
Q: "Simulate $N=25$ trajectories of the Ito Process X satisfying the following S... | The increment of Brownian motion $B_{t+ \Delta }- B_t$ is normally distributed with mean $0$ and standard deviation $\sqrt{\Delta}.$
Generate a sample path using the discrete Euler approximation:
$$X_{k+1}=X_{k} + \mu X_k \Delta + \sigma X_k\sqrt{\Delta}\xi\,\,(k=1,2,...),$$
where $\xi$ is a random number with a standa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Volume of the cooking pot
A cooking pot has a spherical bottom, while the upper part is a truncated cone. Its vertical cross-section is shown in the figure.If the volume of food increases by 15% during cooking, what is the maximum initial volume of food that can be cooked without spoiling ?
It is clear that we have to... | So first consider the bottom spherical part. We know that the distance in the y (vertical) direction from the top of the spherical part to the bottom is 20cm and that the distance across is 40. It should be easy to see that this implies that the radius is 20cm and that we are dealing with a half-sphere. Thus, we can us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$ I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is
$$
\int_0^\pi {\sin\left(n\theta\right)\over
\cos\left(\theta\right) - \cos\left(\alpha\right)}... | Thanks to complex analysis, it is rather easy to obtain
$$
\int_0^{\pi}{\cos\left(n\theta\right)\over
\cos\left(\theta\right) - \cos\left(\alpha\right)}\,{\rm d}\theta=\frac{\pi cos(n\alpha)}{sin(\alpha)}
$$
By the way, this result is also obtained in attachment, but with a method much more complicated than usual. In f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Help to evaluate this limit $\lim_{x \to \infty}x^{\frac{1}{x}}$ What is the value of this limit?
$$
\lim_{x \to \infty}x^{\frac{1}{x}}
$$
I have never encountered such a limit before, so any help or advice would be much appreciated.
| An approach similar to G Tony Jacobs: use the continuity of logarithm (i.e. $\log \lim f(x) = \lim \log f(x)$) to log the expression to get
$$
L f(x) = \frac{\log x}{x}
$$
then show it converges to $0$ by L'Hospital's rule, then exponentiate back to get 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/851227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$ How to prove $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty.$$
I try to do like $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{n+m=N}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{m=1}^{N-1... | $$
\begin{align}
\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m^2+n^2}
&\ge\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m^2+n^2+2mn+m+n}\\
&=\sum_{m=1}^\infty\sum_{n=1}^\infty\left(\frac1{m+n}-\frac1{m+n+1}\right)\\
&=\sum_{m=1}^\infty\frac1{m+1}\\[6pt]
&=\infty
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/851302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 0
} |
Exercise with matrix A) For which $a,b$ is the matrix $A=\begin{bmatrix}
a & 0\\
b & b
\end{bmatrix}$ invertible?
B) Calculate $A^{1000}$ where $A$ is the above matrix with $a=1$ and $b=2$.
$$$$
I have done the following:
A) $$\begin{vmatrix}
a & 0\\
b & b
\end{vmatrix} \neq 0 \Rightarrow ab \neq 0 \Rightarrow a \neq... | Your matrix is diagonalizable, because u have 2 different eigenvalues $\lambda_1=1$ and $\lambda_2=2$. Then you find an invertible matrix $S$, such that $SAS^{-1}=D$ with a diagonal matrix D, which contains your eingenvalues:
Now we obtain the following:
$A^{1000}=(S^{-1}DS)^{1000}=S^{-1}DS*S^{-1}DS*....*S^{-1}DS=S^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Finding the range and domain of $f(x)=\tan (x)$ I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I have done so far.
Finding the domain of $f(x)=\tan(x)$
Consider $f(x)=... | It is easiest to use the intermediate value theorem when finding the range : You know that
$$
\lim_{x\to \pi/2} \tan(x) = +\infty \text{ and } \lim_{x\to -\pi/2} \tan(x) = -\infty
$$
So the image of the interval $(-\pi/2, \pi/2)$ must be $\mathbb{R}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/851424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Power iteration If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration
Especially, I often see that we demand that the matrix is symmetric? Is this necessary?
What seems to be indespensable is that there is a larg... | Let $$M:=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix},\ \text{then}\quad M \begin{pmatrix}1\\ 1 \end{pmatrix}=1\cdot \begin{pmatrix}1\\ 1 \end{pmatrix},\ \text{and} \quad M \begin{pmatrix}1\\ -1 \end{pmatrix}=-1\cdot \begin{pmatrix}1\\ -1 \end{pmatrix} $$
Now let $x^0 =(x^0_1,x^0_2) \neq (0,0)$, then the sequence $x^{k+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why set of natural numbers is infinite, while each natural number is finite? In his book Analysis Vol. 1, author Terence Tao argues that while each natural number is finite, the set of natural numbers is infinite (though has not defined what infinite means yet). Using Peano Axiom, if a property holds for P(0) and whene... | The fact that the infinitude of the natural numbers is effectively taken as an axiom in set theory illustrates to some degree the difficulty in proving that there are infinitely many natural numbers. And you are correct to observe that "infinite" needs to be correctly defined before it makes sense to claim the natural... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 10,
"answer_id": 5
} |
An Integration Calculation I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) which can be found at this link where the following is stated
$$\int_{r_+}^{\infty}dr[f|\psi'|^2+... | Adding more intermediate steps:
\begin{align*}
\int_{r_+}^{\infty}dr[\omega\bar{\psi}\psi'+\bar{\omega}\bar{\psi}'\psi]&=
\int_{r_+}^{\infty}dr[\omega\bar{\psi}\psi'-\bar{\omega}\bar{\psi}\psi'+\bar{\omega}\bar{\psi}\psi'+\bar{\omega}\bar{\psi}'\psi]=\\
&=\int_{r_+}^{\infty}dr[\color{red}{\omega\bar{\psi}\psi'-\bar{\om... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to get $(\frac{x^2}{2}+\frac{1}{2x^2})^2$ from $1+(\frac{x^2}{2}-\frac{1}{2x^2})^2$? How can I get $(\frac{x^2}{2}+\frac{1}{2x^2})^2$ from $1+(\frac{x^2}{2}-\frac{1}{2x^2})^2$?
The book lists the former as the solution to that step.
This is part of an arc length problem, and I think I'm just hitting a mental roadbl... | $$\begin{eqnarray*}
1 + \left(\frac{x^2}{2} - \frac{1}{2x^2}\right)^2
&=& 1 + \left(\frac{x^2}{2}\right)^2 - 2\left(\frac{x^2}{2} \cdot \frac{1}{2x^2}\right) + \left(\frac{1}{2x^2}\right)^2 \\
&=& 1 + \left(\frac{x^2}{2}\right)^2 - \frac{1}{2} + \left(\frac{1}{2x^2}\right)^2 \\
&=& \left(\frac{x^2}{2}\right)^2 + \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $ \lim_{x\rightarrow \infty}\left\{2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right\}$
Evaluate the limit
$$
\lim_{x\rightarrow \infty}\left(2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right)
$$
My Attempt:
To simplify notation, let $A = \left(\sqrt[3]{x^3+x^2+1}\right)$ and $B ... | $a+b=\dfrac{a^3+b^3}{a^2-ab+b^2}$
I think this identity can be used to simplify your expression.
Let $a=\sqrt[3]{x^3+x^2+1}$ and $b=\sqrt[3]{x^3-x^2+1}.$
Then
$a+b=\dfrac{(x^3+x^2+1)+(x^3-x^2+1)}{(x^3+x^2+1)^{2/3}-(x^3+x^2+1)^{1/3}(x^3+x^2+1)^{1/3}+(x^3+x^2+1)^{2/3}}\\=\dfrac{2(x+1/x^2)}{(1+1/x+1/x^3)^{2/3}-(1+1/x+1/x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $ The question is pretty much in the title.
My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$
is convex on $\mathbb{R}\back... | The function $t\mapsto\frac1{1+t}$ is convex on $t\geqslant0$ hence
$$
E\left(\frac1{1+X^2}\right)\geqslant\frac1{1+E(X^2)}=\frac1{1+\nu}.
$$
The lower bound is attained when $P(X=\sqrt\nu)=P(X=-\sqrt\nu)=\frac12$.
On the other hand, if $P(X=0)=1-\frac\nu{x^2}$ and $P(X=x)=P(X=-x)=\frac\nu{2x^2}$ for some $|x|\geqslan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
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