Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to find the following integration Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess the answer should be $\Phi(\mu)$. Here is how I started. Note that $Y:= \bar X$ i... | Find $E[\Phi(c\bar X)]$ where $X_i \sim N(\mu, 1)$ and $c$ is a constant. Then, proceed following Dilip but replace $X$ with $\bar Xc.$
Let $Z\sim N(0,1)$ be independent of all $X_i.$ Then
$$\begin{align}
E[\Phi(c\bar X)]
&= \int_{-\infty}^\infty P\{Z \leq cy\mid \bar X = y\}f_\bar X(y)\,\mathrm dy\\
&= P\{Z \leq c\bar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do you actually calculate inverse $\sin, \cos, $ etc. ? I started to wonder, how does one actually calculate the $\arcsin, \arccos, $ etc. without a calculator? For example I know that:
$$\arccos(0.3) = 72.54239688^{\circ}$$
by an online calculator, but how would one calculate this with a pencil and paper? How is ... | Basically you can use infinite series to calculate approximation of inverse trigonometric functions.
$$
\arcsin z = z+ \left( \frac 12 \right) {z^3 \over 3} + \left( {1 \cdot 3 \over 2 \cdot 4} \right){z^5 \over 5} + \left( {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6} \right){z^7 \over 7}\ +\ ... \; = \sum_{n=0}^\infty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What is the difference between two statements of $\varepsilon-N$ definition? Here is a homework question, TRUE/FALSE:
$$\lim_{n\to\infty}a_n=a\Longleftrightarrow$$
*
*$\forall\varepsilon>0,\ \exists N\in\mathbb{Z^+},\ \text{whenever}\ n>N\Rightarrow|a_n-a|<\varepsilon$. Answer: TRUE
*$\exists N\in\mathbb{Z^+},\fora... | The first is the correct statement. It says: no matter how small $\epsilon$ is, you can always choose a large enough $N$ so that every $a_n$ is within $\epsilon$ of $a$ whenever $n>N$. The second statement says something quite different. It says that if I hand you some large integer $N$, then no matter how small $\epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
How do I verify that $\sin (\theta)$ and $\cos (\theta)$ are functions? I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ a... | The proof is based simply on similar triangles. If a right-angled triangle has an angle $\theta$ then the other two angles are $90^{\circ}$ and $(90-\theta)^{\circ}$. If two triangles have the same angles then they are similar.
My picture shows two similar triangles: $\triangle OAB$ and $\triangle OA'B'$.
Since $\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
Questions for first year students at the University. I will help teach in a introductory class in mathematics for engineers in applied math at the University.
Anyone have any good and cool favorite questions or know where I can find some?
Anything is welcome
For the moment there is some questions like
*
*write: $1 +... | Try asking students to solve for $x$ in the equation $x+\sin(x)=0$.
The solution $x=0$ seems trivial, but it can only be solved by numerical methods, a course of which engineering majors will benefit greatly from.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/820575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Arithmetic and geometric sequences: where does their name come from? Where does the name of these two famous types of sequences come from?
The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it is ... |
Who coined these names? And why?
Who is already very difficult to know. Why is nearly impossible. Because there was no name for the mean?
Anyway, according to a book by Anthony Lo Bello (1), "arithmetic" comes from the Greek word ἀριθμός arithmos, meaning "number". In a similar way, "geometric" comes from γεωμετρία ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 1,
"answer_id": 0
} |
'Obvious' theorems that are actually false It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).
But what are some theorems (prefe... | It's not exactly a theorem, but it fools every math newcomer:
$e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$
$(1 + 1/\infty)$ is $1$, obviously. And 1 to the power of $\infty$ is obviously still 1.
Nope, it's 2.718...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/820686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "371",
"answer_count": 71,
"answer_id": 28
} |
'Obvious' theorems that are actually false It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).
But what are some theorems (prefe... | What about this:
$\mathbb{R}$ and $\mathbb{R}^2$ are not isomorphic (as Abelian
groups with addition).
It falls under the category of "Let's take the Hamel basis of $\mathbb{R}$...", but I like it a lot.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/820686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "371",
"answer_count": 71,
"answer_id": 60
} |
Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$ If $f:\mathbb N\to\mathbb Z$ satisfies:
$$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$
How to show that this implies:
$$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$
I was almost incidentally able to prove this by classifying such functio... | This is actually pretty easy.
Let $n>m$, and take an $N$ such that $N(n-m)>m$. Set $a=N(n-m)-m$. Then
$$
m+a=N(n-m),\qquad n+a=m+a+n-m=(N+1)(n-m).
$$
Now
$$
f(n)-f(m)=f(n)+f(a)-(f(m)+f(a)),
$$
but by assumption $n-m\mid f(m)+f(a)$ and $n-m\mid f(n)+f(a)$, and we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/820731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
In the Quadratic Formula, what does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$? Concerning the Quadratic Formula:
What does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$?
| $\Delta={ b^2-4ac}>0$ means that the equation has two real solutions.
$\Delta= { b^2-4ac}<0$ means that the equation has no real solutions, but two complex solutions.
$\Delta={ b^2-4ac}=0$ means that the equation has one solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/821058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Subsets of intervals If S $\subseteq$ $\mathbb{R}$ is a nonempty, bounded set, and I := [inf S, sup S], show that S $\subseteq$ I. Moreover, if J is any closed bounded interval containing S, show that I $\subseteq$ J. To show that S $\subseteq$ I, let x $\in$ S. Since S is bounded, inf S $\lt$ x < sup S. Thus x $\i... | It's not true that $\inf S < x$; for example, if $S = [0, 1]$ and $x = 0$, this is false. What is true, and follows directly from the definitions, is that
$$\inf S \le x \le \sup S$$
since $\inf$ and $\sup$ are actually bounds on $S$. The result follows immediately from this observation.
For your second question: Sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find integral $\underbrace{\int\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}}}_{n}dx,x>-2$ Find the integral
$$\int\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}}}_{n}dx,x>-2$$
where $n$ define the number of the square
I know this if
$0 \le x\le 2$, then let $$x=2\cos{t},0\le t\le\dfrac{\pi}{2}$$
so... | You were too timid: For $-2\leq x\leq2$ use the substitution
$$x=2\cos t\qquad(-\pi\leq t\leq 0)\ .$$
Then everything goes through as before:
$$\sqrt{2+x}=\sqrt{2+2\cos t}=2\cos{t\over2},\quad \sqrt{2+\sqrt{2+x}}=\sqrt{2+\cos{t\over2}}=2\cos{t\over4}\ ,$$
etcetera.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/821337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Regarding Similarity of matrices Consider a set $T$ of square matrices over finite field $\mathbb{F_p}$. Clearly the cardinality of the set $T$ is $p^{n^2}$ where the square matrices are of size $n$.
Question is: How many non-similar ($A=P^{-1}.B.P$ for some $P$ $\in$ $T$) matrices we would have?
Next, suppose there ... | To your second question: no, the minimal polynomial does not suffice. As an example, consider the following two matrices
$$
A = \begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{bmatrix},
\qquad
B = \begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
\end{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find $x^2 - x$? I'm quite a novice when it comes to maths. I'm on a problem in which I have had to isolate $x$ , through factorials which I completed without problem. However, now I am stuck on a seemingly more minor problem.
The problem I currently have is $x^2 - x = 380$. I know that this can be solved for $x ... | $$x^2 - x = 380$$
Rewrite this as:
$$x^2-x-380=0$$
Use quadratic formula to solve it now. It is:
$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
For:
$$ax^2+bx+c=0$$
Just plug this in:
$$x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-380)}}{2(1)}=\frac{1 \pm \sqrt{1521}}{2}=\frac{1 \pm 39}{2}$$
Which means:
$$\therefore x=20 \text{ or } ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 4
} |
Prove an expression for angle bisector
Show that the vector $\dfrac{A\,\vec B+B\,\vec A}{A+B}$ represents the bisector of the angle between $\vec A$ and $\vec B$.
I can prove that the numerator is the bisector of both vectors but I am unsure how to show that the expression given is as well. Does it matter that the ex... | You are right that the denominator is not that important. Here it serves to give a convex combination of the points $\vec A$ and $\vec B$, i.e., the bisector that is in the segment between these points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/821724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
show than this PDE can be reduced to heat equation How to reduce this PDE to heat equation
$$x^2G_{xx}=G_t$$
($G_{xx}$ is the 2nd order derivative on $x$, $G_t$ is the 1st derivative on $t$)
We wish to obtain a form such that $G(x,t)=F(U(x,t))$,
when substituted into the original equation we have
$$U_{xx}=U_t$$
| Let $\begin{cases}x_1=\ln x\\t_1=t\end{cases}$ ,
Then $G_x=G_{x_1}(x_1)_x+G_{t_1}(t_1)_x=\dfrac{G_{x_1}}{x}=e^{-x_1}G_{x_1}$
$G_{xx}=(e^{-x_1}G_{x_1})_x=(e^{-x_1}G_{x_1})_{x_1}(x_1)_x+(e^{-x_1}G_{x_1})_{t_1}(t_1)_x=(e^{-x_1}G_{x_1x_1}-e^{-x_1}G_{x_1})e^{-x_1}=e^{-2x_1}G_{x_1x_1}-e^{-2x_1}G_{x_1}$
$G_t=G_{x_1}(x_1)_t+G_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving a structure is a field? Please help with what I am doing wrong here. It has been awhile since Ive been in school and need some help. The question is:
Let $F$ be a field and let $G=F\times F$. Define operations of addition and multiplication on $G$ by setting $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)*(c,d)=(ac,db)$. Do... | It can help proving some more general result; let $A$ be any non empty set with an operation $\%$ on it (I use a generic symbol). Consider now the operation $?$ on $A\times A$ defined by
$$
(a,b)\mathbin{?}(c,d)=(a\mathbin{\%}c,b\mathbin{\%}d)
$$
*
*If $\%$ is associative, then also $?$ is associative
*$?$ is commu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are square matrices
Show that if $A,B \in M_{n \times n}(K)$, where $K=\mathbb{R}, \mathbb{C}$, then the matrices $AB$ and $BA$ have same eigenvalues.
I do that like this:
let $\lambda$ be the eigenvalue of $B$ and $v\neq 0$
$ABv=A\lambda v=\lambda Av=BAv$
the third equ... | Alternative proof #1:
If $n\times n$ matrices $X$ and $Y$ are such that $\mathrm{tr}(X^k)=\mathrm{tr}(Y^k)$ for $k=1,\ldots,n$, then $X$ and $Y$ have the same eigenvalues.
See, e.g., this question.
Using $\mathrm{tr}(UV)=\mathrm{tr}(VU)$, it is easy to see that
$$
\mathrm{tr}[(AB)^k]=\mathrm{tr}(\underbrace{ABAB\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 4
} |
Is it sufficient to prove collatz conjecture doing it for $3+6k, k \geq 0$? Thinking about this problem, I saw two interesting properties of Collatz graph. Firstly, if we consider that every even number $e$ can be represented (on a single way) as $e = o 2^n$, where $o$ is an odd number and $n$ is an integer ($n \geq 1$... | Did you see this?
In a previous article, we reduced the unsolved problem of the convergence of
Collatz sequences, to convergence of Collatz sequences of odd numbers, that are
divisible by 3. In this article, we further reduce this set to odd numbers that are congruent to $21\bmod24$…
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Rig categories concept which is equivalent to monoid concept in monoidal categories In monoidal categories, there is a notion of monoid. Is there an "equivalent" concept in rig categories (i.e., categories with two monoidal structures which are related like + and * in a rig)?
| The question is not really precise. And you certainly don't look for an equivalent concept. Perhaps you are looking for the notion of a rig object internal to a rig category? This is an object $x$ equipped with morphisms $z : 0 \to x$ (zero) , $a : x \oplus x \to x$ (addition), $u : 1 \to x$ (unit) and $m : x \otimes x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$ Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but I'm not sure how far that will get you.
| Let $A=QR$ be the QR factorisation of $A$ with $R=[r_1,\ldots,r_n]$. Then Mr Hadamard says that
$$
\left|\,\det A\,\right|
=
\left|\,\det QR\,\right|
=
\left|\,\det Q\,\right|\;\left|\,\det R\,\right|
=
\left|\,\det R\,\right|
=
\prod_{i=1}^n\left|\,r_{ii}\,\right|
\leq
\prod_{i=1}^n\|r_i\|=\prod_{i=1}^n\|a_i\|.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Normal Operators: Numerical Range Disclaimer:
As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.!
Prove that for normal operators the spectrum is contained in the closure of the numerical range:
$$\sigma(N)\subseteq\overl... | If the distance from $\lambda$ to $\mathcal{W}(A)$ is $d > 0$, then
$$
|((A-\lambda I)\phi,\phi)| \ge d\|\phi\|^{2},\;\;\; \phi \in \mathcal{D}(A)\\
\implies d\|\phi\| \le \|(A-\lambda I)\phi\|,\;\;\; \phi \in\mathcal{D}(A).
$$
If $(A\phi,\phi)=(\phi,A^{\star}\phi)$ for $\phi$ in a core dom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What jobs in Mathematics are always in demand, and are deeply Mathematically specialised or greatly general? I am wondering what jobs in the field of Mathematics are (seemingly) always in demand. I am also wondering what jobs there are that are (once again seemingly) greatly Mathematically demanding in regards to eithe... | With the arrive of big data, Statistics, as a branch of Mathematics, is now in high demand. And the challenges there are in many areas. Buzz words like Machine Learning, Analytics, Computer Vision and so on have statistics at their core.
Personally, being a software engineer with a maths background, I recently engaged ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
A difficult integral evaluation problem How do I compute the integration for $a>0$,
$$
\int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx?
$$
I want to find a complex function and integrate by the residue theorem.
| Since it hasn't been specifically objected to yet, here is a solution that doesn't rely on complex variable methods.
We shall make use of the Fourier sine series,
$$\frac{a\sin x}{1-2a\cos x+a^2}=\begin{cases}
\sum_{n=1}^{\infty}a^{n}\sin{(nx)},~~~\text{for }|a|<1,\\
\sum_{n=1}^{\infty}\frac{\sin{(nx)}}{a^{n}},~~~\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
What can we say about the convergence of this series $$\sum {z^n\over n!} $$
I used Alembert's Ration test and get $$\lim_{n \infty}{u_n\over u_{n+1}}={n+1\over z}$$ As this tends to $\infty>1$ can i say that the given series is convergent for all values of $z$ ?
Note : z is a complex number
| Yes. A good thing to remind is the proof of this property, which in your case writes:
$$
\frac{u_{n+1}}{u_n} = \frac z{n+1}
\implies
\left|\frac{u_{n+1}}{u_n}\right| \le 1/2 <1
$$when $n$ is big enough, and then
$$
|u_n| \le C2^{-n}
$$for every $n$, for a certain $C>0$. Hence the series is (absolutely) convergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
infinite discrete subspace In topological space $(X, \tau )$ every compact subspace of $X$ is closed, so no infinite subspace of $X$ can have the cofinite topology.
Is it right to say:
Each infinite subspace of $X$ contains an infinite discrete subspace.
How can I prove it?
Thank you.
| A nice "folklore" topology theorem that is often useful, see the paper: Minimal Infinite Topological Spaces, John Ginsburg and Bill Sands, The American Mathematical Monthly Vol. 86, No. 7 (Aug. - Sep., 1979), pp. 574-576.
Suppose $X$ is any infinite topological space. Then there exists a countably infinite subspace $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find all functions $f$ such that $f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)$.
Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)\text.$$
I have derived these clues:
*
*$f(0)=0$;
*$f(x^2)=xf(x)$;
*$f(x)=-f(-x)$.
But now I am confused. I know solution will ... | I don't know how to do this without continuity. If you can show $f(x) \to f(0)$ whenever $x \to 0$, then the original expression yields it. Then all you need is $f(x)$ bounded near zero in order to use $f(x^2) = x f(x)$ to get that the limit exists and is zero.
However, taking continuity for granted, consider your resu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Prove $a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$ increasing There is a homework question in Calculus-1 course:
Calculate the limit of $\{a_n\}$: $$a_1=1,\ a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$$
I think the key points are bounded and increasing, and I have proved that $$a_n\in(1, 2)$$
If I knew it's increasing then $$a=1+\frac{a}{1+... | First, we prove that $a_n$ is bounded above. Obviously, $a_n = 1+ \frac{a_{n-1}}{a_{n-1}+1} < 2$.
To prove that the sequence is increasing, we can just use induction. The base cases are trivial so suppose the claim holds for all naturals $ \le k$. Then $a_{k+1}-a_k = \frac{a_k}{1+a_k} - \frac{a_{k-1}}{1+a_{k-1}} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$L$-function of an elliptic curve and isomorphism class Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$.
If two elliptic curves are isomorphic, they clearly have the same $L$-function.
What about the converse ? If two elliptic curves $E... | Isogenous.
A theorem of Faltings says that two elliptic curves $E_1$ and $E_2$ over a number field $F$ are isogenous if and only if they have the same $L$-factors at almost all places. See, for example, this article, Theorem 3.1, or find the source:
G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpe... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Inverse Z-transform with a complex root The z-transform of a signal is
$$ X(z)=\frac{1}{z^2+z+1}$$
I attempted to solve for the the inverse z-transform by decomposing the denominator into complex roots, $\alpha$ and $\alpha^\ast$, to get
$$\frac{1}{z^2+z+1} = \frac{A}{z-\alpha}+\frac{B}{z-\alpha^\ast}=\frac{\frac{-i\... | Notice that $(1-z)(1+z+z^2)=1-z^3$. (That's a trick worth noting whenever you're working with $1+z+z^2+\ldots z^n$.)
So $1/(z^2+z+1)$ is actually $(1-z)/(1-z^3)$.
$1/(1-z^3)$ is the sum of a geometric progression ie. $1+z^3+z^6+\ldots$.
So the final result is $1+z^3+z^6+\ldots-z(1+z^3+z^6+\ldots)$
So if we write $X(z)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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a question about the evaluation of triple integral, I am stuck! How to use the method of orthogonal transformation to figure out the triple
integral ?. I am stuck about it! The triple integral is:
$$
\iiint\cos\left(ax + by + cz\right)\,{\rm d}x\,{\rm d}y\,{\rm d}z
\qquad\mbox{and}\qquad x^{2} + y^{2} + z^{2} \leq 1
$$... | Let $\vec{u}$ be the vector $(a,b,c)$.
Let $\lambda = |\vec{u}| = \sqrt{a^2+b^2+c^2}$ and
$\displaystyle\;\hat{u} = \frac{\vec{u}}{|\vec{u}|}$ be the associated unit vector.
Pick two more unit vectors $\hat{v}$, $\hat{w}$ such that
$\hat{u}, \hat{v}, \hat{w}$ forms an orthonormal basis. You then parametrize the points... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Volume between cylinder and plane Problem: Find the volume bounded by $z = y^2, x =0, y =0, z =9-x$.
My working:
$z$ goes from $y^2$ to $9-x$ so these are the limits of integration.
Work out the points of intersection of $9-x$ and $y^2$. When $y=0$, $9-x=0$ and $x=9$. So $x$ goes from 0 to 9. When $x=0$, $y^2 = 9$ so $... | Your limits of integration don't make sense. The region of integration is given by the set $$R = \{(x,y,z) \in \mathbb R^3 \mid (y^2 \le z \le 9-x) \cap (0 \le x \le 9) \cap (0 \le y \le 3)\}.$$ The projection of $R$ onto the $xz$-plane is simply the triangle $x \ge 0$, $z \ge 0$, $x + z \le 9$. Over this triangle, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to reason about two points on the unit sphere. I've recently been thinking about various problems involving two points on the surface of a unit sphere. Let's specify them with a pair of unit 3-vectors ${\bf \hat a}$ an ${\bf \hat b}$. Is there some aid to thinking about this 4-dimensional space? In particular:
... | This is $S^2 \times S^2$; you can give it the product topology, so that a basis for the topology is given by $\{U \times V \mid U, V \text{ open } \subset S^2\}$.
In order to think about 1 and 2, it might help to think of it as a manifold; you can use polar coordinates to homeomorphically biject open sets to open sets ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Similarity between matrices. I have two matrices,
$A = \begin{pmatrix} 1&2\\3&4\end{pmatrix}$
$B =\begin{pmatrix} 1&2\\-1&-4\end{pmatrix}$
I need to check if A is similar to B.
I did by first computing the characterstic polynomial of the first one and the second one.
I got that $\det(t I-A) = t^2 -5t -2$
and $\det(t I-... | Now, let's prove similar matrices have equal traces.
General proposition:
Let $A\in M_{mn}(\mathbb F)$ and $B\in M_{nm}(\mathbb F)$. Then $\operatorname{trace}(AB)=\operatorname{trace}(BA)$.
Note: $AB\in M_m(\mathbb F)$ and $BA\in M_n(\mathbb F)$.
As stated in previous answers,
two matrices $A,B\in M_n(\mathbb F)$ ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How find the approximate $\ln{2}$ such the error is less than $0.001$ if $$1.4142<\sqrt{2}<1.4143$$, use it to approximate
$$\ln{2}$$ such the error is less than $0.001$
This is National Higher Education Entrance Examination.
| Let us try : $$\log(2)=2\log(\sqrt 2)$$ Now, let us use $$\log\frac{1+x}{1-x}=2\Big(\frac{x}{1}+\frac{x^3}{3}+\frac{x^5}{5}+...\Big)$$ and use $$\frac{1+x}{1-x}=\sqrt 2$$ that is to say $x=3 - 2 \sqrt 2$, which means, according to $1.4142<\sqrt{2}<1.4143$, $0.1714<x<0.1716$ (this is a small number which will make the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$ $$\int_0^1 \frac{\arcsin(x)}{x}dx$$
This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma functions, but I can't even find a way to start it. Wolfram Alpha gives a kilometric so... | Let $y=\arcsin x\;\Rightarrow\;\sin y =x\;\Rightarrow\;\cos y\ dy=dx$, then
$$
\int_0^1 \frac{\arcsin(x)}{x}dx=\int_0^{\Large\frac\pi2}y\cot y\ dy.
$$
Now use IBP by taking $u=y$ and $dv=\cot y\ dy\;\Rightarrow\;v=\ln(\sin x)$, then
\begin{align}
\int_0^{\Large\frac\pi2}y\ \cot y\ dy&=\left.y\ln(\sin y)\right|_0^{\Larg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this integral with sine and cosine such a challenge? ...or maybe I just don't know some specific trick with trigonometric functions?
Well, anyway, here it is:
$$\int{\sin^6{x}\cos^4{x}\, dx}$$
I'm bored with it, because I get 9 integrals out of 1 and the whole thing frustrates me as hell.
So, is there a simpler w... | How about this?
$$\int\sin^6x\cos^4xdx=\int\sin^4x\cos^4x(\sin^2x)dx=\frac1{32}\int\sin^42x(1-\cos2x)dx=$$
$$\frac1{32}\int\sin^42xdx-\frac1{32}\int\sin^42x\cos2xdx$$
This should cut your work down quite a bit.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and no idea about $\sum\limits_{n=1}^\infty \dfrac{(3+4i)^n}{5^n\,\sqrt[999]{n}}$ How can one see that $ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and by which criterion? I was using a binomial theorem for $ (1+i)^n $ as $ \sum\limits_{k=... | This is for your second sum:
$$|3+4i|=5$$
$$\arg(3+4i)=\arctan(4/3)=:\alpha \approx 1$$
$$\sum\limits_{n=1}^\infty \frac{(3+4i)^n}{5^n\,\sqrt[999]{n}} =
\sum\limits_{n=1}^\infty \frac{\exp(\alpha n i)}{\sqrt[999]{n}}$$
$$\frac{1}{\sqrt[999]{n}} \downarrow 0$$
I'm pretty much certain that this sum will converge by some... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear map from zero vector to zero vector. I am reading an introduction on linear maps in my text book on linear algebra. The following statements are made:
Suppose $G_1 (\vec{u}) = (x_1 + 2x_2 + 3x_3 + 1, 4x_1, 9x_3)$
Then we can use the following property of linear maps. Let $\lambda = 0$ and $\vec{u} = \vec{0}$
$$G... | You are being confused. There are two statements going on here.
The first statement is a property of linear transformations. Specifically, it is true that $G(\lambda \mathbf{u}) = \lambda G(\mathbf{u})$ for any $\lambda$ and any $\mathbf{u}$.
A consequence of this is that every linear map must map $\mathbf{0}$ to $\mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to explain the perpendicularity of two lines to a High School student? Today I was teaching my friend from High School about linear functions. One of the exercises we had to do was finding equations of perpendicular and parallel lines. Explaining parallel equations was quite easy, if we have the equation $y = ax + ... | Give a good answer to this question may be also a way to give a rigorous definition of "perpendicular" and of what is the "measure" (in radians) of an angle (a not simple task at elementary level) without using angles measured in degrees.
I sketch the reasoning in steps:
1) Let's $O$ the intersection point of two str... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Equation of matrices Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that:
$$[ T ]_E = \left( \matrix{
0 & 0 & a \cr
1 & 0 & b \cr
0 & 1 & c
} \right)$$
where $a,b,c \in F$.
Show there's $v\in V$ such that:
$$v \in \ker\left(T^3 ... | This seems to be a case of Cayley's theorem ; the result that every matrix is a root of its characteristic polynomial . Compute the characteristic polynomial and then, by Cayley's theorem, if you evaluate $T$ in it (with $T^n=T \circ T\circ\cdots\circ T$
n times ) you will get $0$ (meaning the matrix with all-zero entr... | {
"language": "en",
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Determine the Cumulative Distributive Distribution(CDF) of a truncated value?
It is the last part(part h) that I am having problems with.
I know you use integration and then split it into 2 parts.
But how exactly do you do it ? A detailed answer would be very helpful !
Please help !
| It is a lot simpler than you might think. Here is a hint: there are two cases if $U = \min\{Y, b\}$. Either $Y < b$ or $Y \ge b$. If the latter, then $\Pr[U = b] = \Pr[Y \ge b] = S_Y(b)$. Otherwise, $Y < b$ implies $U = Y$ and you already know the density for that situation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integral in $\mathbb{R}^n$ What the value of $$\int\limits_{|x|\geq1}\frac{1}{|x|^n}dx , \quad x \in \mathbb{R}^n?$$ This integral is part of other problem.
| The integral
$$
K(n,\alpha) := \int\limits_{|x|\geq1}\frac{1}{|x|^\alpha}dx , \quad x \in \mathbb{R}^n?
$$
converges only if $\alpha > n$. But as far as I know there is no known closed form. Perhaps there are asymptotics as $\alpha \downarrow n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does a row of zeros make a free variable in linear systems of equations? I don't understand how a row of zeros gives a free variable when solving systems of linear equations. Here's an example matrix and let us say that we're trying to solve Ax=0:
$$\left[
\begin{matrix}
2 & -3 & 0 \\
3 & 5 ... | Fewer equations than unknowns means you cannot solve the set in a unique way. If you have three variables and in effect two equations as you have, then any one of the three variables can be seen as free. However, once you choose a value for it, the value of the two other variables will be forced.
Studying interactions... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Possibilities Of Dividing Cards In a card game we give 13 cards for each of the 4 players.
How much division of card are there?
I thought it is ${4\choose1}*{52\choose 13}$, but the answer is $\frac{52!}{13!^4}$
where did I get wrong?
| This would be multinomal coefficient. We have 52 cards to seperate to 4 players, each of size 13. So:
$\dbinom{52}{13,13,13,13}$
Your way would be to count how many ways can you hand 13 cards out of 52 cards to one of four players, which is different than what the question asks for.
| {
"language": "en",
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What is an example of real application of cubic equations? I didn't yet encounter to a case that need to be solved by cubic equations (degree three) !
May you give me some information about the branches of science or criterion deal with such nature ?
| To expand on @DrkVenom's post, elliptic curve cryptography (ECC) is a great example of the deep application of cubics. ECC is currently at the forefront of research in public key cryptography. It has the important benefit of requiring a smaller key size while maintaining strong security. Here is an accessible and wel... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
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Every weakly convergent sequence is bounded Theorem: Every weakly convergent sequence in X is bounded.
Let $\{x_n\}$ be a weakly convergent sequence in X. Let $T_n \in X^{**}$ be defined by $T_n(\ell) = \ell(x_n)$ for all $\ell \in X^*$. Fix an $\ell \in X^*$. For any $n \in \mathbb{N}$, since the sequence $\{\ell(x_n)... | The equality $\|x_n\|=\|T_n\|$ is an instance of the fact that the canonical embedding into the second dual is an isometry.
See also Weak convergence implies uniform boundedness which is stated for $L^p$ but the proof works for all Banach spaces.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
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"answer_id": 0
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Approximate the probability An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it travels 10cm in a random direction, forwards or backwards on the rubber band. Every night, the rubber band gets stretched uniformly by 1km. As an example, during the first day the snail might advance... | I do not believe the snail will reach the end in finite time, but it may reach the end in the limit, Assuming the extreme case, the snail always moves forward, at time 0, the snail moves 10cm to the right, and is at 10cm on a 1km band. The band then stretches, so the snail is now at 20cm on a 2km band. The snail moves ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 2,
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$2$-dimensional Noetherian integrally closed domains are Cohen-Macaulay Any 1-dimensional Noetherian domain is Cohen-Macaulay (C-M).
For the $2$-dimensional case, a condition of being integrally closed is necessary to be added for a Noetherian domain to be C-M, which I could not prove it.
Would anybody be so kind a... |
Let $R$ be a noetherian integral domain of dimension $2$. If $R$ is integrally closed, then $R$ is Cohen-Macaulay.
From Serre's normality criterion we have that $R$ satisfies $(R_1)$ and $(S_2)$.
$(R_1)$ gives that all the localizations of $R$ at height one primes are regular, and therefore Cohen-Macaulay. (In fact, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What are the terms for the elements in the Euclidean algorithm $a = qb + r$? In a ring $R$ with a Euclidean norm $N$, given $a,b \in R$ with $b\neq 0$, there are elements $q, r$ such that $a = qb + r$.
Is there any special terminology for the elements $a,b,q,r$ in this context? For example, if we write $a = \frac{p}{q}... | $a = qb + r$
*
*$q$ quotient,
*$b$ divisor,
*$r$ remainder,
*$a$ dividend.
| {
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"url": "https://math.stackexchange.com/questions/826010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding an equation of circle which passes through three points How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.
I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is... |
I know i need to use that formula but have no idea how to start
\begin{equation*}
\left( x-q\right) ^{2}+\left( y-p\right) ^{2}=r^{2}\tag{0}
\end{equation*}
A possible very elementary way is to use this formula thrice, one for each point. Since the circle passes through the point $(5,10)$, it satisfies $(0)$, i.e.
$$... | {
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"url": "https://math.stackexchange.com/questions/827072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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$\lim_{x \to 0} \frac{e^{\sin2x}-e^{\sin x}}{x}$ without L'Hospital Anyone has an idea how to find $\displaystyle\lim_{x \to 0} \dfrac{e^{\sin2x}-e^{\sin x}}{x}$ without L'Hospital? I solved it with L'Hospital and the result is $1$ but the assignment is to find it without L'Hospital. Any idea?
| $e^{sin2x} \sim_0 sin2x + 1$, and also $e^{sinx} \sim_0 sinx + 1$. Thus:
$\dfrac{e^{sin2x} - e^{sinx}}{x} \sim_0 \dfrac{sin2x + 1 - sinx - 1}{x} \sim_0 \dfrac{2x - x}{x} = \dfrac{x}{x} = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/827287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Solving a simple matrix polynomial Does there exist a $2\times 2$ Matrix $A$ such that
$A-A^2=\begin{bmatrix} 3 & 1\\1 & 4\end{bmatrix}$ ?
| Yes, there are complex solutions, but no real solutions. Using the brute force method also mentioned by JimmyK, you can see that $A$ must be symmetric. The top left element then becomes $a-a^2-b^2$ which must equal $3$, which can obviously only happen if $A$ has complex elements.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you prove that $ n^5$ is congruent to $ n$ mod 10? How do you prove that $n^5 \equiv n\pmod {10}$ Hint given was- Fermat little theorem. Kindly help me out. This is applicable to all positive integers $n$
| By Fermat's Little Theorem, $n^5 \equiv n \pmod 5$
Also $n \equiv 0 \ or \ 1 \pmod 2 \implies n^5 \equiv n \pmod 2$
Chinese Remainder Theorem guarantees the presence of a solution $\pmod {10}$ as $(2,5) = 1$
From the second congruence, $n^5 = 2k + n$
Substitute into the first congruence, $2k+n \equiv n \pmod 5 \implies... | {
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How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? I know how to expand $\frac{1}{\sqrt{1-x}}$, but I have no idea how to expand $\frac{1}{\sqrt{1-x^2}}$. Simply differentiate this makes expanding way to complicated.
| Simply substitute $x^2$ for $x$ in the expansion of$(1-x)^{-1/2}$
| {
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Is $\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$? For matrices $A,B\in\mathbb{R}^{m\times n}$ and for any unit vector $x$, is the following true, and if so why?
$\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$
Equivalently, is $x$ the eigenvector of $A^TA - B^TB$ correspon... | $$\lVert Ax\rVert^2-\lVert Bx\rVert^2=x^T(A^TA-B^TB)x\le \lVert A^TA-B^TB\rVert $$ if $x$ is a unite vector. This is true if $A^TA-B^TB$ is positive semidefinite.
Now, since $A^TA-B^TB$ is symmetric it is diagonalizable unitarily and hence $A^TA-B^TB=U\Lambda U^T$ for some orthogonal (unitary) matrix $U$. Then, the exp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fourier series of coshx using fourier of $e^{x}$. I have to find the Fourier series of $coshx$ on $(-l,l)$.What I did was I found the Fourier series of $e^{x}=\sum _{n=-\infty}^{\infty }{(-1)^n (\ell^2+in\pi)\over{l^2+n^2\pi^2}}\sinh(\ell)e^{{in\pi x}\over\ell}$
Since $\cosh x={e^{x}+e^{-x} \over 2}$ ,to find $e^{-... | Yes, you can. Let's do it more carefully: introduce a new variable $t$, assigning to it the value $t=-x$. Then
$$e^{-t}=\sum _{n=-\infty}^{\infty }{(-1)^n (\ell^2+in\pi)\over{l^2+n^2\pi^2}}\sinh(\ell)e^{{-in\pi t}\over\ell} \tag1$$
But what's in a name? The formula (1) is valid, no matter what the variable is called.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.
Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.
My attempt :
f is $2\pi$ periodic and $\mathcal{C}^3$, we have :
$$\lim_{h\rightarrow 0, h>0} \frac{f(h)-f(0)}{h}=\lim_{h\rightarrow... | A possible direction:
Write out the Fourier series:
$$f(x)= a_0 + \sum\left( a_n \cos nx + b_n \sin nx\right)$$
$$g(x)\equiv f'(x)+f'''(x) = -\sum n (n^2-1) \left(b_n \cos (n x)-a_n \sin (n x)\right)$$
Clearly, when $n=1$, $g(x)=0$, meaning that the lowest order in $g(x)$ is $n=2$. Now, since there is no constant term,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/827969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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If $I_n=\sqrt[n]{\int _a^b f^n(x) dx}$,for $n\ge 1$.Find with proof $\lim _{n \to \infty}I_n$ Let $a,b \in \mathbb R ,a<b$ and let $f:[a,b] \to [0,\infty) $ a continuous and non constant.
attempt,using Reimann series
$I_n=\sqrt [n]{\int _a^b f^n(x) dx}$
$I_n=\sqrt [n]{\lim _{k \to \infty}\sum_{i=1}^k f^n(x_i^*) dx}$... | Let $\epsilon > 0$.
1. As $f(x)\le \sup f$, $$\int_a^b f^n(x) dx \le (b-a)(\sup f)^n
\\ \limsup \left(\int_a^b f^n(x) dx\right)^{1/n} \le \limsup (b-a)^{1/n}\sup f = \sup f
$$
2. There are $a'<b'$ such as $a'<y<b' \implies f(y) > \sup f-\epsilon$, because $f$ is continuous.
$$
\liminf \left(\int_a^b f^n(x) dx\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Matrices and rank inequality Let $A \in K^{m\times n}$ and $B \in K^{n \times r}$
Prove that min$\{rk(A),rk(B)\}\geq rk(AB)\geq rk(A)+rk(B)-n$
My attempt at a solution:
$(1)$ $AB=(AB_1|...|AB_j|...|AB_r)$ ($B_j$ is the j-th column of $B$), I don't know if the following statement is correct: the columns of $AB$ are a li... | Use the dimension theorem on $A|_{\mathop{\rm Im}(B)}:\mathop{\rm Im}(B)\subseteq K^n\longrightarrow K^m$. Then $$\begin{align}\dim(\mathop{\rm Im}(B))&=\dim(\mathop{\rm Im}(A|_{\mathop{\rm Im}(B)}))+\dim(\mathop{\rm Ker}(A|_{\mathop{\rm Im}(B)}))\\&=\dim(\mathop{\rm Im}(AB))+\dim(\mathop{\rm Ker}(A)\cap\mathop{\rm Im}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Concerning a specific family of recursive sequences There is a family of recursive sequences that I came up with: $$a_n=\text{sum of factors less than $a_{n-1}$ of } a_{n-1}, a_0\in\mathbb{N}$$ First question: Has it been studied before?
Here are some observations I've made about it:
Starting from successive positive ... | The sequence in question is known as aliquot sequence. Based on the references I was able to find, question #1 and #3 are open problems at the moment; we don't even know whether :
*
*the sequence starting with $276$ ever becomes periodic.
*there is any sociable number of order $3$ (number whose aliquot sequence has... | {
"language": "en",
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Show that $\vert\int_{-1}^1 \omega(t) dt \vert \leq 2^n \int_{-1}^1\vert \omega^{(n)}(t)\vert dt$ I am stuck with the following problem:
With $\omega: [-1,1]\rightarrow \mathbb{R}$, $\omega\in C^n(-1,1)$. Suppose that $\omega$ has a finite number of zeroes $t_1<t_2<\cdots <t_n$ (i.e. $\omega(t_i)=0,\forall i$) on $[-... | Applying Rolle's theorem an awful lot of times, you will find that all derivatives of $\omega$ of orders $\le n-1$ have at least one zero in $[-1,1]$. This allows you to run the following, outrageously wasteful, chain of inequalities for $k=0,1,\dots, n-1 $:
$$\int_{-1}^1 |\omega^{(k)}(x)|\,dx \le 2 \sup_{[-1,1]} |\om... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve this problem using either Ptolemy's Theorem or Law of Cosines? A hexagon is inscribed in a circle of radius r. Suppose that four of the edges of the hexagon are 10 feet long and two of the edges are 20 feet long, but the exact arrangement of the edges is unknown. What is the value of r to three decimal pla... |
Consider the following diagram where I have rearranged the sides such that $EF = 20, FC = ED = 10, DC = 2r$ and $A$ is the center.
Let $H$ be the altitude to $DC$ from $F$. Note that $\angle DFC = \angle AGF - \angle DFG + \angle HFC$. However, since $CF' = DE = 10, \angle DFG = \angle HFC$ so $\angle DFC = 90^\circ$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there a term for an algebraic structure with two binary operators that are closed under a set? For example, let's say we're using the operators +, and *, and the set {0,1,2}
The Cayley tables look like this:
* 0 1 2 + 0 1 2
0 0 0 1 0 1 2 0
1 1 2 1 1 0 1 0
2 0 0 2 2 1 2 2
These Cayley tables are totally ... | From Burris, Sankappanavar A Course in Universal Algebra page 26 (42 of the pdf): "An algebra A is unary if all of its operations are unary, and it is mono-unary if it has just one unary operation."
Although from what I read it is not clear whether or not in practice this terminology has been extended before, an algebr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Limits with factorial I'm having difficulties understanding all limits with factorial...
Actually, what I don't understand is not the limit concept but how to simplify factorial...
Example :
$$\lim\limits_{n \to \infty} \frac{(n+1)!((n+1)^2 + 1)}{(n^2+1)(n+2)!}$$
I know that it's supposed to give $0$ as I have the an... | $\frac{(n+1)!}{(n+2)!} = \frac{1}{n+2}$
So, the problem reduces to $lim_{n\rightarrow \infty} \frac{(n+1)^2 + 1)}{(n+2)(n^2 + 1)}$.
The numerator is quadratic in $n$ while the denominator is cubic, so as $n \rightarrow \infty$ the limit should go to $0$.
| {
"language": "en",
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Help to compute sum of products I need to compute the following sum:
$$(1\times2\times3)+(2\times3\times4)+(3\times4\times5)+ ...+(20\times21\times22)$$
All that I have deduced is:
*
*Each term is divisible by $6$. So sum is is divisible by $6$.
*Sum is divisible by $5$ as 1st term is $1$ less than multiple of $5$ ... | Here's an interesting solution:
$(1\cdot2\cdot3)+(2\cdot3\cdot4)+(3\cdot4\cdot5)+\dots+(20\cdot21\cdot22)$
$\dfrac{3!}{0!}+\dfrac{4!}{1!}+\dfrac{5!}{2!}+\dots+\dfrac{22!}{19!}$
$3!\left(\dfrac{3!}{0!3!}+\dfrac{4!}{1!3!}+\dfrac{5!}{2!3!}+\dots+\dfrac{22!}{19!3!}\right)$
$3!\left(\dbinom{3}{3}+\dbinom{4}{3}+\dbinom{5}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $\sqrt[3] {2} ,\sqrt[3] {4},1$ are linearly independent over rationals I was trying to prove that $\sqrt[3] {2} ,\sqrt[3] {4}$ and $1$ are linearly independent using elementary knowledge of rational numbers. I also saw this which was in a way close to the question I was thinking about. But I could not com... | Consider $c_1\sqrt{2}+c_2\sqrt{3}+c_3\sqrt{5}=0$. Then $c_1\sqrt{2}+c_2\sqrt{3}=-c_3\sqrt{5}$. Squaring both sides we will have $2c_1^2+3c_2^2+2\sqrt{6}c_1c_2=5c_3^2$. If either $c_1$ or $c_2$ turns out to be $0$ then we will either have $c_2\sqrt{3}+c_3\sqrt{5}=0$ implying $3c_2^2=5c_3^2$ which gives $\left(\frac{c_2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove infinite series $$
\frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2}
$$
I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be $x>1$, the question doesn't specify. Putting the LHS into Wolfram Alpha doesn't generate the RHS (it... | I think a less formal solution could be more understandable.
consider $$ S_n= \frac{1}{x} + \frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \dots + \frac{n}{x^n}$$
$$ xS_n = 1 + \frac{2}{x} + \frac{3}{x^2} + \frac{4}{x^3} + \dots + \frac{n}{x^{n-1}}$$
then
$$xS_n - S_n = 1+ (\frac{2}{x}-\frac{1}{x})+(\frac{3}{x^2}-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Conditional Probability | Please explain this answers Can you please explain Why P(W2) is 4/9 ?
Thanks..
| This is because you are searching for the white marbles and there are total 4 white marbles. And the total marbles you've got is 9. So it 4/9.
You don't actually need this method, You just need the probability of the second draw.Because the second draw is without replacement, Total marbles decrease by 1 and now the tot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829349",
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Why does $\sum_{k=1}^{\infty}\frac{{\sin(k)}}{k}={\frac{\pi-1}{2}}$? Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$.
Presumably, this can be derived from the similarity of the Leibniz expansi... | $\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\dow... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove the following Trig Identity with reciprocals Prove that:
$$\frac{\tan x}{\sec x-1}+\frac{1-\cos x}{\sin x}=2\csc x$$
Help please! I tried so many things but couldn't get the LHS = RHS. A hint please?
| Hint:
$$
(1 - \cos x)(1 + \cos x) = 1 - \cos^2 x = \sin^2 x\\
(\sec x - 1)(\sec x + 1) = \sec^2 x - 1 = \tan^2 x
$$
Further
$$
\frac{\sec x + 1}{\tan x} = \frac{1+\cos x}{\sin x} = \frac{(1 + \cos x)^2}{(1 + \cos x) \sin x}\\
\frac{\sin x}{1 + \cos x} = \frac{\sin^2 x}{(1 + \cos x) \sin x}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/829622",
"timestamp": "2023-03-29T00:00:00",
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Shortest path between wikipedia articles I'm trying to figure out whether it is possible (and if so how) to find the shortest path inside a network from one node to another. I know that there are different possible algorithms to do that the most prominent being probably the A* search algorithm. I know that this algorit... | Thanks for your answers! I kept looking for solutions and found something very useful on stackoverflow. The answer is actually exactly what I expected. What I need is some function which makes an prediction for every node that is connected to the current on how long (relatively speaking) the total path will be. The pos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fitch-Style Proof Hi I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.
Premises:
$A \land (B \lor C)$
$B \to D$
$C \to E$
Goal: $\neg E \to D$
Thank You
| You should be able to transform the following in a formal proof.
Assume $\neg E$.
Prove $B\lor \neg B$ with the intent to use $\lor$-$\text{Elim}$.
If $B$ holds, then use $\to$-$\text{Elim}$ on the premise $B\to D$ to conclude $D$.
Suppose $\neg B$ holds. Use $\land$-$\text{Elim}$ on the first premise to get $B\lor C$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is $(1 - \frac{1}{p})^n$ close to $e^{-\frac{n}{p}}$ when $n$ and $p$ are large? Looking at this answer by Henry birthday problem - expected number of collisions and struggling to figure out why it matches this other formula provided to me on a programming related question. Thanks!
| I don't know if you are familiar with Taylor series, but if yes :
$$\left(1-\frac1p\right)^n=\exp\left(n\ln\left(1-\frac1p\right)\right)=\exp\left(-n\left(\frac1p+o\left(\frac1p\right)\right)\right)$$
and this gives :
$$\left(1-\frac1p\right)^n\sim_{p>>1} \exp\left(-\frac{n}{p}\right)$$
Note than there's no assumption ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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(Elementary) Trigonometric inequality Any idea for proving the following inequality: $5+8\cos x+4 \cos 2x+ \cos3x\geq 0$ for all real x? I've tried trigonometric identities to make squares appear, and other tricks; but nothing has worked well.
| Note that $\cos 2x = 2\cos^2 x - 1$ and $\cos 3x = 4\cos^3 x - 3\cos x$.
So, we want to prove that $1+5\cos x + 8\cos^2x + 4\cos^3 x \ge 0$.
The left side is a polynomial in $\cos x$ which can be factored as $(1+\cos x)(1+2\cos x)^2$.
Can you take it from here?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When does the limit of the mean values of a function around a point approach the value of the function at that point ? When does the limit of the mean values of a function around a point
approach the value of the function at that point ?
We can prove it if the function is
continuous. But are there general classes of ... | The answer to this is the content of the Lebesgue differentiation theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
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Find all the asymptote of $1-x+\sqrt{2+2x+x^2}$ I find
$$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}=2,$$ but i'am stuck when $x\rightarrow-\infty$ how to find that $y=-2x$ is an oblique asymptote. Any idea?
| You can use the asymptotic expressions for $f(x)$:
$$f(x) \sim 2 +\frac{1}{2x} + O(x^{-2}) \quad \text{as} \quad x \rightarrow \infty$$
$$f(x) \sim -2x -\frac{1}{2x} + O(x^{-2}) \quad \text{as}\quad x \rightarrow - \infty$$
and read-off the asymptotes $y=2$ and $y=-2x$.
PS: To get the second expression consider
$$g(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/830229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Infinite sum of logarithms Is there any closed form for this expression
$$
\sum_{n=0}^\infty\ln(n+x)
$$
| As others correctly mentioned, the expression diverges. Yet, if necessary, you can get quite good asymptotics:
$$
\sum_{k=1}^{n} \log (k+x) = \sum_{k=1}^{n} \log k + \sum_{k=1}^{n} \log (1+ \frac{x}{k}) \sim n \log n + \sum_{k=1}^{n} \frac{x}{k} = n \log n + x \log n \\
=(n+x) \log n
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/830295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is the integral of the arc length in polar form not similar to the length of the arc of a circular sector? So I learned that the area enclosed by a polar function is computed by
$$A = \int \frac{r(\theta)^2}{2}d\theta.$$
Which, I learned, comes somewhat from the formula for the area of a circular sector
$$A_{sector... | You have the same phenomenon in rectangular coordinates:
The area under a curve $y=f(x)$ $\> (a\leq x\leq b)$ is given by the integral
$$\int_a^b f(x)\ dx\ ,$$
which "comes somewhat" from the formula for the area of a rectangle
$$A_{\rm rectangle}= {\rm height}\cdot{\rm width}\ .$$
So one could expect that the integral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/830364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Hessian matrix to establish convexity I have a function, $u(x_1,x_2)=\alpha \ln(x_1)+(1-\alpha)\ln(x_2)$.
where $0<\alpha <1$
I want to prove that it is convex.
The Hessian matrix I have constructed is:
$$ \left( \begin{array}{ccc} -\alpha/x_1^2 & 0 \\ 0 & -(1-\alpha)/x_2^2)\end{array} \right)$$
From here I found that ... | If $\alpha$ is real than either $\alpha>0$ or $1-\alpha>0$ and if $0<\alpha<1$ then both are positive. If $\alpha>0$, consider the two points $(x_1,x_2)$ and $(w_1,x_2)$. If $x_1$, $w_1$ are distinct positive numbers then we would have
$$
u(p(x_1,x_2) + (1-p)(w_1,x_2)) > pu(x_1,x_2)+(1-p)u(w_1,x_2).
$$
If by "convex"... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y
Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y are natural)
... | For the first question, any square number modulo four is 0 or 1 [even ==> 0, odd ==> 1]. This can be shown rather easily. Then, the summation of two such numbers would never equal 3 modulo four showing the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/830577",
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Geometrically integral Here is a stupid question about the notion of geometric integrality.
Say I have a smooth, projective variety $X$ over a some field $k$, equipped with a morphism $f: X \to C$ to a smooth, projective curve $C$, such that the generic fibre is geometrically integral.
Assume that there exists a finite... | Yes, absolutely. The generic fiber $Y$ is an algebraic variety over the function field $K$ of $C$. Consider $K$ as a finite non-trivial extension of a subfield $L$, then $Y \times_L (K \otimes_L \overline{L})$, and $K \otimes_L \overline{L}$ is never integral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/830663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\sum_{n = 1}^\infty f_n =f.$ Prove that $f′_n =f′$ a.e. I am studying for a real analysis qualifying exam. Was hoping that there was a very slick proof for this? Thanks.
Let $f_1, f_2, . . . , f : [0, 1] → \mathbb{R}$ be non-decreasing right-continuous functions such that $\sum_{n = 1}^\infty
f_n =f.$ Prove that $\s... | You can write $f_n(x)=\mu_n([0,x])$ for some nonnegative measure $\mu_n$. Write that as $d\mu_n=f'_n\,d\lambda+d\nu_n$, where $\lambda$ is Lebesgue measure and $\nu_n$ is singular. Do the same for $f$: $d\mu=f'\,d\lambda+d\nu$. Let $N$ be a (Borel) null set where $\nu_n([0,1]\setminus N)=0$ for all $n$, and likewise $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/830752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How is math used in computer graphics? I'm doing a research paper on the mathematics of computer graphics and animation (3D) and I do not know where to start. What mathematical equations and concepts are used for computer graphics and animation (3D)?
| Two main results that tend to form the core of 3D graphics:
(1) 3D coordinates are represented by matrices:
$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$
Transforms (like rotations) are represented by matrix multiplication:
$$\begin{bmatrix} \text{new-x} \\ \text{new-x} \\ \text{new-x} \end{bmatrix}
= \begin{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/830856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Partial derivatives-Why does this stand? In my notes there is the following:
$$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix}
u_{\xi}=0 \Rightarrow u=g(\eta)\\
u_{\eta}=0 \Rightarrow u=f(\xi)
\end{matrix}\right.$$
I haven't understood why this stand...
Isn't it as followed??
$$u_{\xi \eta}=0 \Rightarrow u_{\xi}=F(\... | You are correct. $u_\xi=F(\xi),\ u_\eta = G(\eta)$. Note that this implies (by integrating) that $u(\xi,\eta) = \widetilde F(\xi) + \widetilde G(\eta)$. You can verify this by trying something simple like $u(\xi,\eta) =\xi+\eta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/830962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Difference between Euclidean space and vector space? I often hear them used interchangeably ... they are very complicated to make any use of.
Wikipedia words:
Euclidean space:
One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.
Ve... | While a vector space is something very formal and axiomatic, Euclidean space has not a unified meaning. Usually, it refers to something where you have points, lines, can measure angles and distances and the Euclidean axioms are satisfied. Sometimes it is identified with $\mathbb{R}^2$ resp. $\mathbb{R}^n$ but more as a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 6,
"answer_id": 3
} |
Paradox of the trumpet shape This is a question I had for long time now, when you rotate the function $y=1/x$, $x>0$ (say $x$ and $y$ both measure meters) about the $x$ axes by $2\pi$ you get a shape which has infinite surface area and finite volume.Lets call this shape "trumpet shape". Now the weird thing is that supp... | Your issue is trying to compare a 2 dimensional object (surface area) with a 3 dimensional object (volume). Any volume of liquid can be spread thin enough to cover as much surface area as you want (mathematically speaking, there are probably physical limits to this).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/831208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Show that $1$ is an eigenvalue of $A$ Let $A \in \mathbb{K}_{n}$. Suppose that $A^3=I_{n}$ and $I_{n}+A+A^2\neq O_{n}$. Where $O_{n}$ is the null matrix. Show that $1$ is an eigenvalue of $A$.
I couldn't show what is ask by using all the hypothesis. My work was:
Let $v \in \mathbb{K}_{nx1}$, and $\lambda \in \mathbb{K}... | Assume by contradiction that 1 is not an eigenvalue of $A$. Then $\det(A-I) \neq 0$, and therefore $A-I$ is invertible.
Hence
$$A^2+A+I=(A^3-I)(A-I)^{-1}=0$$
contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/831290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
if a function $f$ is decreasing and the limit $\lim\limits_{x\to \infty} f(x)$ exists, then Given that function $f$ is decreasing and the limit $\lim\limits_{x\to \infty} f(x)$ exists
How can I prove $$\lim\limits_{x\to \infty} x\left(f(x)-f(x+1)\right)$$ exists?
I applied monotone convergence theorem, but there is no ... | This is false. Counterexample:
Define $f(x)$ this way, for each positive integer $n$ then for all $n^{2}<x\leq (n+1)^{2}$ then $f(x)=-\sum\limits_{i=1}^{n}\frac{1}{i^{2}}$. Clearly $f(x)$ is bounded below (bounded by $-\frac{\pi^{2}}{6}$) and monotone decreasing.
Thus whenever $x=(n+1)^{2}$ then $x(f(x+1)-f(x))=x(-\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Generator of singular homology of n-sphere I am learning singular homology theory right now. The homology of n-sphere is computed by Mayer-Vietoris argument. Intuitively, for example the class represented by a loop is the generator of $H_1(S^1)$. Is there a way to show that this is truly the case in singular homology? ... | You have a long exact sequence
$$\dots→\tilde H_1(A)\oplus \tilde H_1(B)→\tilde H_1(A+B)→\tilde H_0(A∩B)→\tilde H_0(A)⊕\tilde H_0(B)→\dots$$ where $H_n(A+B)$ is the homology group for the chain complex
$$\dots→C_n(A+B)→C_{n-1}(A+B)→\dots$$
where $C_n(A+B)$ consists of chains whose simplices are each in $A$ or in $B$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Application of Riemann Roch I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than $g+1$. Unfortunately I have found only very abstract references which not clearly implies this f... | Let $D$ be a finite collection of $d$ points on $\Sigma$. The RR theorem shows
that the vector space of meromorphic functions on $\Sigma$ with at worst a simple pole at each $D$ has dimension $\geq d + 1 - g$, which is $> 1$ if $d > g$. Thus (choosing any $D$ with $d > g$) this space contains a non-constant meromorp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$ I have a question:
$$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$$
Thanks for your help>
| Since $|-2|<3$ then $(-2)^n=_\infty o(3^n)$ and then
$$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=\lim_{n \to \infty} \frac{3^n}{3^{n+1}}=\frac13$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/831731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Towers of Hanoi recurrence relation
How would I do this recurrence relation?
| Let T(n) be the number of moves needed to transfer n disks from one peg to another.
Clearly we have:
T(0)=0
T(1)=1
Now, we note that in order to move a tower of n disks, we need to do the following:
Move the tower of n−1 disks from off the top of the nth disk onto another of the pegs;
Move the nth disk to the destinati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/831814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is the property reflexive, symmetric, anti-symmetric, transitive, equivalence relation, partially ordered given the relation below? I'm working on this and I'm supposed to figure out if the following properties apply to the below relations.
Properties are:
1. Reflexive
2. Symmetric
3. Anti-Symmetric
4. Transitive
5. Eq... | If I were correcting your (presumably) homework, I would want more details on your reasoning for transitivity, in both cases. Nevertheless, all your answers are correct. Good job.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/831936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$f(x) = x^3 - x$ then $f(n)$ is multiple of 3 If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$.
First I tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$
When $x$ is an integer then at least one factor on the right is even, and exactly one factor on the right is divisible by $3$. It follows that... | After you get $f(n)=n(n+1)(n-1)$, meaning $f(n)$ has factors $n$, $n+1$, $n-1$. Now you want to show $3$ divides one of them.
If $3\mid n$ then you get what you want.
If $3\nmid n$, then $n\equiv 1$ or $n\equiv 2 \pmod 3$. Meaning $3\mid n-1$ or $3\mid n+1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Intro to Real Analysis I am having trouble proving the following:
if $a < b$, then $a < {a+b\over2} < b$.
I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where to go from there.
Any suggestions?
| Given $a<b$,
$$a<b \Longleftrightarrow 2a<a+b \Longleftrightarrow a<\frac{a+b}{2}$$
$$a<b \Longleftrightarrow a+b<2b \Longleftrightarrow \frac{a+b}{2} < b$$
$$a<\frac{a+b}{2}<b$$
as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 6
} |
Symbol for rational/irrational part of a number Just as $\Im(z)$ and $\Re(z)$ denote the imaginary and real parts of $z$, respectively, do there exist symbols for the rational and irrational parts of a real number?
| I think the closest thing you can get is the floor function. Where the "rational" or integer part of $x$ would be the largest integer less than or equal to $x$. But this doesn't really guarantee that what is left over will be irrational.
Edit
Thinking about it a bit more I think the following is at least well defined ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
What was the book that opened your mind to the beauty of mathematics? Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But this question is specifically about which books inspi... | W. W. Sawyer's 'Prelude to Mathematics' is a great book that really opened my eyes. It can be read almost without any knowledge of mathematics.
I also believe that any mathematician ought to read 'Flatland'. It is a beautiful story it gave me my fist real intuition about higher dimensions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 51,
"answer_id": 6
} |
What was the book that opened your mind to the beauty of mathematics? Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But this question is specifically about which books inspi... | Apostol's Introduction to Analytic Number Theory. It's a beautifully written and self contained book. Even if you cannot solve all the problems, just reading the text will take you a long way. One of the best number theory books I've seen.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 51,
"answer_id": 38
} |
Closest Point to a vector in a subspace Given v = [0 -3 -8 3], find the closest point to v in the subspace W spanned by [6 6 6 -1] and [6 5 -1 60]. This is web homework problem and I have used the formula (DotProduct(v, w.1)/DotProduct(w.1, w.1))*w.1 + (DotProduct(v, w.2)/DotProduct(w.2, w.2))*w.2 but the computer said... | The projection of a vector $v$ over $\mathrm{span}(w_1, w_2)$ is the sum of the projections of $v$ over $w_1$ and $w_2$, if $w_1$ and $w_2$ are orthogonal. If they're not, you can find other vector that span the same subspace, using Gram-Schmidt's process. For example: $$\mathrm{proj}_{w_1} v = \frac{\langle v, w_1\ran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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