Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding a differentiable function that cannot be bounded by quadratic function Is there any function from $\mathbb R$ to $\mathbb R$ such that it's value at $0$ is $0$ and its derivative at $0$ is also $0$ and can never be bounded by any form of quadratic function in any neighborhood of the origin?
I have been thinkin... | What about $f(x) = |x|^{3/2}$?
It satisfies your condition, but $\lim_{x\to 0} \dfrac{f(x)}{x^2} = \infty$, so $f(x)$ is larger than $cx^2$ near $x=0$ for every $c>0$.
(You may have another kind of "bound" in mind though.)
| {
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"url": "https://math.stackexchange.com/questions/1731921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Evaluate $\Re[\cos(1+i)]$ Evaluate $\Re[\cos(1+i)]$.
The trigonometric function in the expression is throwing me in a loop and need some guidance on how to evaluate this.
Thanks.
| $$2\cos(a+ib)=e^{i(a+ib)}+e^{-i(a+ib)}=e^{-b}e^{ia}+e^be^{-ia}$$
Use Euler Identity: $e^{ix}=\cos x+i\sin x$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Convergence of a Branching process Consider the Branching process: $\{ \xi_i^n , n \ge 1, i \ge 1\}$ are i.i.d. taking values $0, 1, \ldots$ and $Z_0 := 1, \; Z_{n+1} := \sum\limits_{i=1}^{Z_n} \xi_i^{n+1}$. Assume $\mu := \mathbb{E}[\xi_i^b]>1$. Assume $\sigma^2 := \operatorname{Var}(\xi_i^n) < \infty$. Denote $X_n :=... | Suppose the conditional expectation of $X_n$ given $X_{n-1}$ is $X_{n-1}$, as in a martingale, and the conditional variance is $2$. Then the law of total variance tells us that
$$
\operatorname{var}(|X_n - X_{n-1}|^2) = \operatorname{var}(\operatorname{E}(|X_n-X_{n-1}|^2 \mid X_{n-1}) + \operatorname{E}(\operatorname{... | {
"language": "en",
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If $M$ is a connected manifold, does $M\setminus\{p\}$ have finitely many components?
Let $M$ be a connected manifold and $p\in M$. Is it true that $M\setminus\{p\}$ has only finitely many connected components?
(We can also suppose $M$ is compact if that helps.)
I think this is true but I can't prove it yet. This is ... | I think your argument is quite all right. The crucial part of it is proving the implication $M - \{p\}$ has infinitely many components $\implies$ $V - \{x\}$ has infinitely many components, and I think you should focus on making sure it you argue it convincingly.
Note though that if $\dim M \geq 2$, $M - \{p\}$ is conn... | {
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"url": "https://math.stackexchange.com/questions/1732583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Neumann condition for Poisson equation Solving $ \nabla^2u = 1 $ for spherically symmetric u in the region $r < a, a > 0$, with the following conditions at r = a (separately)
(a) $u = 0$
(b) $\nabla u \cdot n = 0 $ where n is the outward normal of the region.
So generally $ u(r) = 1/6 r^2 + Ar^{-1} + B $ where A,B ar... | Note that a necessary condition for the existence of a solution to the Neumann problem is that
$$\begin{align}\int_{r\le a} (1)\,dV&=\int_{r\le a}\nabla^2 u(r)\,dV\\\\
&=\oint_{r=a}\nabla u(r)\cdot \hat n\,dS \tag 1
\end{align}$$
The left-hand side of $(1)$ is simply $\frac{4\pi a^3}{3}$ while if $u'(a)=0$, the right-... | {
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Least involved proof for continuos functions $\Rightarrow$ uniform continuous functions on $[a,b]$ I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is:
If f is continuous on a closed interval [a,b], then f is uniformly continuous on [a,b].
... | Suppose for contradiction that $f$ is continuous but not uniformly continuous on $[a,b]$. Then there exists $\varepsilon > 0$ such that for every $n \in \mathbb{N}$ there exist $x_n, y_n \in [a,b]$ with $|x_n - y_n| < \frac{1}{n}$ and $|f(x_n) - f(y_n)| \ge \varepsilon$. By the Bolzano-Weierstrass theorem, the sequence... | {
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Prove that a homomorphism is injective or trivial Let A,B be groups, and assume that |A| = 29. Let φ:A→B be a homomorphism.
a) Prove that either φ is injective or trivial. (φ is trivial if for all a∈A φ(a) = e)
b) If |B|=80, prove that φ is trivial.
Now I know that a homomorphism is injective iff the kernel is trivial.... | By the first isomorphism theorem, $A/\ker\varphi \cong \varphi(A)$. In particular: \begin{equation}\frac{29}{|\ker\varphi|}=|\varphi(A)|.\end{equation} Since 29 is a prime number, either $\ker\varphi=\{0\}$ (which implies $\varphi$ is injective) or $\ker\varphi=A$ (i.e. $\varphi$ is trivial).
If $|B|=80$ then since $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1732931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The quintic equation: Why is there no closed formula? We know that polynomials up to fourth degree have closed solutions using radicals. And we know that starting from the quintic no polynomial will have a closed solution using radicals.
Question 1: What I want to know is, why does this happen for the fifth order polyn... | If by ‘closed formula’, you mean a formula with radicals and ordinary arithmetic operations, the general answer comes from Galois theory:
One consider the Galois group of the equation, i.e. the group of automorphisms of the splitting field of the polynomial. Suppose the polynomial has degree $n$. As an automorphism of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1733072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $f(x)=\frac{1}{1-x-x^2}$, find $\frac{f^{(10)}(0)}{10!}$ My math teacher posed this question to my calculus class. If $f(x)=\frac{1}{1-x-x^2}$, find $\frac{f^{(10)}(0)}{10!}$.
At first, I began by taking the first few derivatives, but it soon go out of hand with the repeated quotient rules. I'm sure that I could con... | If you don't know the theory of generating functions, the trick is to get the Taylor expansion with partial fractions. Write
$$
\frac{1}{1-x-x^2}=\frac{a}{1-\alpha x}+\frac{b}{1-\beta x}
$$
that gives the relations
$$
\begin{cases}
a+b=1\\[4px]
a\beta+b\alpha=0\\[4px]
\alpha+\beta=1\\[4px]
\alpha\beta=-1
\end{cases}
$$... | {
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Does a finite sum of distinct prime reciprocals always give an irreducible fraction?
If we add any finite number of any distinct prime reciprocals, will the result always be an irreducible fraction?
If not, is there any bound on the value of a greatest common divisor for the numerator and demoninator of such fraction?... | As you've shown, this comes down to the question of whether
$$P=\sum_{k=1}^n p_1p_2\ldots\hat p_k\ldots p_n$$ and $p_1\ldots p_n$ are coprime, where $\hat a$ means that the product excludes $a$.
The only candidates for common prime factors are the $p_i$. But
$$S\equiv p_1p_2\ldots\hat p_i\ldots p_n\pmod {p_i}$$which is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Newton's method for square roots 'jumps' through the continued fraction convergents I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square roots.
Explanation. If we start Newto... | First some experimentation might be in order. I wrote up a program that could check for chains like this:
program nr
implicit none
integer, parameter :: ik16 = selected_int_kind(38)
integer, parameter :: N = 200
integer(ik16) p(N), q(N)
integer D
integer sqD
integer r, s, a
integer i, j, k
in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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The closure of a subset in finer topology is always subset of the closure of that subset in the coarser one. On the Appendix A of Naber's book $\textit{The Geometry of Minkowski Spacetime}$ there is a claim in Lemma A.3.3. It says that if we have a set (says $M$) endowed with two different topology says $(M,O_A)$ and $... | Easier proof: $Cl_B(U)$ is an $A$-closed set which contains $U$, hence $Cl_B(U)$ must also contain $Cl_A(U)$, by definition of $Cl_A$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Supremum/Maximum and Infimum/minimum of a given set Determine $\sup E$, $\inf E$, and (where possible) $\max E$, $\min E$ for the set $E = \{ \sqrt[n]{n}: n \in \mathbb{N}\}$.
Attempt: I've written that $\inf E = 1 = \min E$.
When it comes to finding $\sup E$, I've noticed punching in increasing values of n on my calcu... | The function $f(x)=x^{\frac{1}{x}}$ has derivative
$$ f^{\prime}(x)=x^{\frac{1}{x}}\frac{1-\log x}{x^2}$$
Therefore $f$ has its global maximum on $[1,\infty)$ at $x=e$, and is increasing on $[1,e)$ and decreasing on $(e,\infty)$. Therefore the only values of $n$ you need to check are $n=2$ and $n=3$. And $3^{\frac{1}{3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there an elegant way to solve $\int \frac{(\sin^2(x)\cdot \cos(x))}{\sin(x)+\cos(x)}dx$? The integral is:
$$\int \frac{(\sin^2(x)\cdot \cos(x))}{\sin(x)+\cos(x)}dx$$
I used weierstraß substitution
$$t:=\tan(\frac{x}{2})$$
$$\sin(x)=\frac{2t}{1+t^2}$$
$$\cos(x)=\frac{1-t^2}{1+t^2}$$
$$dx=\frac{2}{1+t^2}dt$$
Got this... | If you multiply numerator and denominator by $\cos x-\sin x$, the numerator can be rewritten as
$$
\sin x\cos x(\sin x\cos x-\sin^2x)
$$
Now use $\sin x\cos x=\frac{1}{2}\sin 2x$ and
$$
\sin^2x=\frac{1-\cos2x}{2}
$$
so finally we get
$$
\frac{1}{4}\sin2x(\sin2x-1+\cos2x)=
\frac{1}{4}(1-\cos^22x-\sin2x+\sin2x\cos2x)
$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Proof that exact form are path independent seems to imply the same for merely closed forms A singular $k$-cube on some set $A \subseteq \mathbb R^n$ is a continuous map $c : [0,1]^k \to A$. Consider the following exercise:
Let $c_1, c_2$ be singular $1$-cubes in $\mathbb R^2$ with $c_1(0) = c_2(0)$ and $c_1(1) = c_2(1... | You are somehow right: the proof would work if you only knew that $\omega$ were closed, provided that $\omega$ were defined not just on $\partial C$, but on all $C$. But in that case, all closed forms on $C$ are exact.
That is not the case on $\mathbb{R}^2 \setminus \{ 0\}$. If you pick $C_1$ and $C_2$ to bound a regi... | {
"language": "en",
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Strengthening Poincaré Recurrence Let $(X, B, \mu, T)$ be a measure preserving system. For any set $B$ of positive measure, $E = \{n \in \Bbb N |\; \mu(B \; \cap \;T^{-n}B) > 0\}$ is syndetic.
This exercise comes from Einseidler and Ward. The exercise before is the "uniform" mean ergodic theorem which is proved basica... | Hint: If the set was not syndetic, then the sequence
$$
\frac1n\sum_{k=0}^{n-1}\mu(B\cap T^{-k}B)
$$
would have zero as an accumulation point (take larger and larger gaps). But
$$
\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\mu(B\cap T^{-k}B)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\int_B(\chi_B\circ T^k)\,d\mu=\int_B\lim_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Nowhere $0$ form on the sphere? Consider the differential form on $\mathbb R^3$ given by $ x dy \wedge dz + y dz \wedge dx + z dx \wedge dy$. I converted this to spherical coordinates using a laborious calculation, and when I'm done, by some miracle (which would be cool if someone could explain exactly how that works),... | First of all, there's just a point on the unit sphere where $\phi=0$ — the north pole (but there's also the south pole, where $\phi=\pi$, to worry about). But remember that spherical coordinates actually fail to give a coordinate system at these points (and we can debate what happens when $\theta = 0$ or $2\pi$).
In th... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Can an alternating series ever be absolutely convergent? Can an alternating series EVER be absolutely convergent?
I am examining practice problems in my calculus book and I haven't yet come across a case where this is so. It might be because they are simple, but I'm genuinely curious.
| a series is absolutely convergent if $\sum |a_n| < M$
If a series is absolutely convergent then every sub-series is convergent.
Consider $\sum (-1)^n|a_n|$
The sum of the of the even terms converges, the sum of the odd terms converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1734249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Eigenvalues within unit circle Let
$$A = \begin{bmatrix} -1 & -N\\ 6 & 0\end{bmatrix}$$
from the state space realization of an LTI system. For this system to be stable, all eigenvalues must be within the unit circle, i.e., for all eigenvalues $|\lambda_i|<1$ must be satisfied. Matrix $A$ has eigenvalues
$$\lambda_{1,2}... | The two roots are
$$
\lambda_1(N) = -\frac{-1+ \sqrt{1-24N}}{2}, \;\;\;
\lambda_2(N) = -\frac{-1- \sqrt{1-24N}}{2}.
$$
It is easily seen that the condition for real roots are $N \le \frac{1}{24}$.
We consider real eigenvalue case first. The real eigenvalues are within the unit disc if
$$
-1 \le \lambda_i(N) \le 1, \;\;... | {
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"timestamp": "2023-03-29T00:00:00",
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Can I approximate a series as an integral to find its limit and determine convergence? Find $\lim \limits_{n \to \infty} (a_n)$, where $a_n=\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+...+\frac{n}{n^2}$.
So I can solve it like that $a_n=\frac{1+2+3+...n}{n^2}=\frac{\frac{1}{2}n(n+1)}{n^2}=\frac{1}{2}(1+\frac{1}{n})$. Cle... | This can be written as
$$\lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^n {r\over n}$$
This is of the form
$$\lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^n f\left ({r\over n} \right) $$
So it can be written as
$$\int_0^1 f(x)dx$$
$$=\int_0^1x dx$$
$$=\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1734494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When is $\sum_{N=1}^{\infty}\exp\left(\ln\left(\frac{\sqrt{N}\ln(N)}{g(N)}\right)-\frac{(g(N))^2}{\ln(N)}\right)<\infty$? Let
\begin{align}
\sum_{N=1}^{\infty}\exp\left(\ln \left(\frac{\sqrt{N}\ln(N)}{g(N)}\right)-\frac{(g(N))^2}{\ln(N)}\right)
\end{align}
Question:
Let $g(N)=a(\ln(N))^{t}$ where $a \geq 0$ is some co... | If $g(N)=a(\ln(\ln(N)))^{t}$, rewrite terms in decreasing asymptotic magnitude and get rid of the insignificant ones.
$\displaystyle \ln \left(\frac{\sqrt{n}\ln(n)}{g(n)}\right)=\frac 12 \ln n+\ln \ln n -t\ln \ln \ln - \ln a$
$\displaystyle \frac{(g(n))^2}{\ln(n)}= \frac{a^2(\ln\ln n)^{2t}}{\ln n}=o(1)$
Hence $$\exp\le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Function that maps the "pureness" of a rational number? By pureness I mean a number that shows how much the numerator and denominator are small.
E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is less pure than the previous examples, $\frac{53}{41}$ is worse, ....... | $f(\frac{a}{b})=\frac{1}{|a|+|b|}$
For the following conditions on $x$, $f(x)$ is either zero or not defined:
*
*$x$ irrational
*$x=0$
Higher output values implies high purity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1734697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Differentiate $\sqrt{1+e^x}$ using the definition of a derivative This is the progress I've made so far.
$$\lim_{h \to 0} \frac{\sqrt{1+e^{x+h}}-\sqrt{1+e^{x}}}{h}$$
$$= \lim_{h \to 0} \frac{\left(1+e^{x+h}\right)-\left(1+e^{x}\right)}{h\left(\sqrt{1+e^{x+h}}+\sqrt{1+e^{x}}\right)}$$
$$= \lim_{h \to 0} \frac{e^x\left(e... | $$\lim_{h \to 0} \frac{e^x}{\left(\sqrt{1+e^{x+h}}+\sqrt{1+e^{x}}\right)} \cdot \frac{\left(e^h-1\right)}{h}$$
Since $$e^h=\frac{h^0}{0!}+\frac{h}{1!}+\frac{h^2}{2!}+\frac{h^3}{3!}+......$$
$$e^h=1+\frac{h}{1!}+\frac{h^2}{2!}+\frac{h^3}{3!}+......$$
$$e^h-1= \frac{h}{1!}+\frac{h^2}{2!}+\frac{h^3}{3!}+......$$
$$\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1734853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find a recurrence expression which solution have $\sin$ or $\cos$. I, I'm a computer science student of the first course.
My teacher have told us to try to find a recurrence equation for the closed-form expression:
$$f(n) = 2^n + 3^n \cos\left(\frac{n\pi}{2}\right) $$
and I think I need a bit of help.
At the first, ... | You are so close! Since $e^{\frac{n\pi}2i}=\cos\left(\frac{n\pi}2\right)+i\sin\left(\frac{n\pi}2\right)$, you need $$e^{\frac{n\pi}2i}=\left(e^{\frac{\pi}2i}\right)^n=\left(\cos\left(\frac{\pi}2\right)+i\sin\left(\frac{\pi}2\right)\right)^n=(0+i)^n=i^n$$
Thus your second term is $3^ni^n=(3i)^n$ so you need the root of ... | {
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Coloring classes of $\{1,2,3,\dots,n\}$ I'm trying to prove the following statement
There is an integer $n_0$ such that for any $n\ge n_0$, in every $9$-coloring of $\{1,2,3,\dots,n\}$, one of the $9$ color classes contains $4$ integers $a,b,c,d$ such that $a+b+c=d$.
I thought about taking $\{1,2,\dots,n\}$ to be ver... | Isn't this a direct consequence of Rado's single equation theorem?
(taken from the book Ramsey Theory over the Integers by B. Landmann and A. Robertson)
where $c_1=c_2=c_3=1$, $c_4=-1$ for your statement and we can take e.g., $D=\{c_1, c_4\}$. By "regular", it means your statement is true for all $r$-colorings with $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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functional, compact operator I am working on a homework problem (Analysis now E3.3.7) and I have no idea on how to solve it. Can anyone give some thoughts? Many thanks.
Assume that Hilbert pace $H$ is separable and prove that an operator $T$ in $B(H)$
has the form $U|T|$ for some unitary operator $U$ with the $|T|=(T^*... | Remaining spaces:
$$\overline{\mathcal{R}|A|}^\perp=\mathcal{N}|A|=\mathcal{N}A\quad\mathcal{N}A^*=\overline{\mathcal{R}A}^\perp$$
For equal dimensions:
$$\dim\mathcal{N}A=\dim\mathcal{N}A^*\implies UU^*=1$$
For more details: Polar Decomposition
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Continuous functions Let $X$ and $Y$ be topological spaces and $f$ a function of $X$ into $Y$. Show that $f$ is continuous if and only if es continuous as a function of $X$ onto the subspace $f(X)$ of $Y$.
I'm proceding like this:
First, assume that $f$ is continuous of $X$ into $Y$, let $f(X) \cap A$ be an open set of... | If you like to be pedantic, one could say this is an immediate consequence to the characterisation of continuity of maps into a space with the initial topology. I.e. if we have $f: X \rightarrow Y$ and $i: f[X] \rightarrow Y$ is the inclusion map, and $\tilde{f}: X \rightarrow f[X]$ is the image restriction of $f$, so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solving $e^x = 6x$ for $x$ without a graph. Throughout my high school career I was always told that an equation of this sort ( $e^x = 6x$ for example) couldn't be solved algebraically. However I feel that there may be a way (and you may be out there saying "of course there is a way") I know that it can be solved graphi... | I am a fan of fixed point iteration because I find it easy to set up. You want to work your equation into the form $x=f(x)$ where the derivative of $f(x)$ at the root is small and certainly less than $1$ in absolute value. Then pick a reasonable starting value for $x_0$ and iterate $x_{i+1}=f(x_i)$. As logs are slow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Prove by strong induction that $3^n$ divides $a_n$ for all integers $n \ge 1$
Let $a_1 = 3, a_2 = 18$, and $a_n = 6a_{n-1} − 9a_{n-2}$ for each integer $n \ge 3$. Prove by strong induction that $3^n$ divides $a_n$ for all integers $n \ge 1$
I've done the base step and ih however I am stuck on the Inductive Step. I'm ... | Use strong induction: assume $a_{n - 2} = 3^{n - 2}\lambda_{n - 2}$ and $a_{n - 1} = 3^{n - 1}\lambda_{n - 1}$. We then have:
\begin{align}
a_n =&\ 2\cdot 3\cdot 3^{n - 1}\lambda_{n - 1} - 3^2\cdot 3^{n - 2}\lambda_{n - 2}\\
=&\ 2\lambda_{n - 1}3^n - \lambda_{n - 2}3^n \\
=&\ \left(2\lambda_{n - 1} - \lambda_{n - 2}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
limit of product exists and one limit exists Question is to check :
If $\lim_{n\rightarrow \infty}a_nb_n$ exists and $\lim_{n\rightarrow \infty}a_n$ exists implies $\lim_{n\rightarrow \infty}b_n$ exists.
Considering $a_n=\frac{1}{n}$ and $b_n=n$ then we see that $\lim_{n\rightarrow \infty}a_nb_n$ exists, equals to $... | If
$$\;\lim_{n\to\infty}a_n=L\neq0\;,\;\;\lim_{n\to\infty}a_nb_n= K\;,\;\;\text{then since for almost all indexes}\;\;a_n\neq0\,,$$
we get that for all indexes except a finite number of them, from arithmetic of limits:
$$b_n=\frac{a_nb_n}{a_n}\xrightarrow[n\to\infty]{}\frac KL$$
and all this is well-defined and always... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$ Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$.
In some textbooks on algebraic number theory I have seen the fact: $\vert N_{K / \mathbf{Q}}(\alph... | I have written up the proof in Lemma $3.3.3$ of my lecture notes in algebraic number theory, on page $35$. It uses three different $\mathbb{Q}$-bases of the number field $K$, $\mathbb{Z}$-bases for the the ring of integers $\mathcal{O}_K$, and the determinant for the commutative diagram given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1735863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Cut-off function construction Let $f:I=[0,1]\cup[2,3]\to \mathbb{R}$, defined by
$$f(x)= \begin{cases} 0 & \text{if } x\in [0,1] \\ 1 & \text{if } x\in [2,3] \end{cases} $$
How do I construct a $C^1$ function $\tilde{f}: \mathbb{R}\to \mathbb{R}$ such that $\tilde{f}(x)=f(x)$ for all $x\in I$.
| I will show you how to fill in the gap $(1,2)$ and leave you to get the rest. The easiest method is to use a polynomial to fill in the gap. It will need to have a derivative of zero at $x=1$ and $x=2$, so the polynomial needs to be at least cubic, and its derivative has the form $$p'(x)=a(x-1)(x-2)=ax^2-3ax+2a,$$ where... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Residue Theorem: compute the integral $\int_0^\infty \frac{x \sin x}{x^4+4a^4}$
Compute the integral $$\int_0^\infty \frac{x \sin x}{x^4+4a^4}$$
Since, it's an even function I can rewrite the expression as $$\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin x}{x^4+4a^4}$$.
In the previous part I found that the integral a... | Write the integral of interest $I(a)$ as
$$\begin{align}
I(a)&=\frac12\int_{-\infty}^\infty \frac{x\sin(x)}{x^4+4a^4}\,dx\\\\
&=\text{Im}\left(\frac12\int_{-\infty}^\infty \frac{xe^{ix}}{x^4+4a^4}\,dx\right) \tag 1\\\\
&=\lim_{R\to \infty}\text{Im}\left(\frac12\oint_{C_R}\frac{ze^{iz}}{z^4+4a^4}\,dz\right)
\tag 2\\\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Function as an eigenvector for a matrix? So I am currently going through this paper: Link Here
And in section 2.2, it defines $K$ to be a weighted adjacency matrix for a certain rectangular $n$ by $m$ graph, where the weights are all either $1$ or $i$. It then goes on to say that for fixed $j$ and $k$, the function:
$$... | It appears that the standard basis of the vector space the linear operator $K$ acts on is most easily enumerated by two indexes $x,y$ where $x\in\{1,\ldots,m\}$ and $y\in\{1,\ldots,n\}$. To specify a vector in this space, we need to specify the component this vector has for each possible combination of $x$ and $y$ from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Efficiently evaluating $\int x^{4}e^{-x}dx$ The integral I am trying to compute is this:
$$\int x^{4}e^{-x}dx$$
I got the right answer but I had to integrate by parts multiple times. Only thing is it took a long time to do the computations. I was wondering whether there are any more efficient ways of computing this int... | Here is a nice little trick to integrate it without using partial integration.
$$
\int x^4 e^{-x} \,\mathrm dx = \left. \frac{\mathrm d^4}{\mathrm d \alpha^4}\int e^{-\alpha x} \,\mathrm dx \right|_{\alpha=1} = \left.- \frac{\mathrm d^4}{\mathrm d \alpha^4} \frac{1}{\alpha} e^{-\alpha x}\right|_{\alpha=1}
$$
The ide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 2
} |
Trouble with dependent matrix solution Im determining the eigenvector for $\lambda = 6$. Here is the following matrix $A - 6*I$:
0 0 0 0
0 1 0 1
0 0 0 0
0 0 0 0
Thus the corresponding equation: $x_2 = -x_4$
Thus $x_1,x_3,x_4$ are free.
How do I express this in terms of an eigenvector, a little confused..?... | A simpler example to show what can happen.
$$\left[\begin{array}{cc}2&1\\0&2\end{array}\right]$$
You will get $(2-\lambda)^2 = 0$ to solve for eigenvalues. So two eigenvalues at $2$.
$$\left[\begin{array}{cc|c}
0&1&0\\
0&0&0
\end{array}\right]$$
Wee see that the only requirement is $x_2 = 0$ and $x_1$ can be what it wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Different names for model with parameters specified and not? Say I have a general model:
$$y=\beta_{1}x_{1}+\beta_{0}$$
or
$$y=\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{0}$$
I might be performing some operations to determine which general model to choose.
Say I have decided on a general model structure and want to define a ... | The coefficients $\beta_1,\beta_2,\beta_0$ in this context are often called parameters. Particular values of the parameters determine a particular member of a parametrized family of models.
In the context of linear regression, one sometimes says those three parameters are fixed (as opposed to random) and unobservable,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Dimension of two subspaces Good evening. I'm trying to show that if the sum of the dimensions of two subspaces of a vector space exceeds the dimension of space then these subspaces have a vector in common. I have trouble to build the proof, because no how is this possible.
| Let $V$ be the vector space, and $W$, $U$ be the two subspaces. Choose bases $\{w_1,\dots,w_m\}$ and $\{u_1,\dots,u_n\}$ for $W$ and $U$, respectively.
If $m+n>\dim V$, then the set $\{w_1,\dots,w_m,u_1,\dots,u_n\}$ is linearly dependent, so there are scalars $c_1,\dots,c_{m+n}$ not all zero such that
$$ c_1w_1+\dots+c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Non-Changing Determinant When Adding (Seemingly) Arbitrary Entries Question:
I've found that adding what seem to be arbitrary values in the 4th row don't change the value of the determinant. Why is that?
A = $\begin{bmatrix}
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 4 & 0 \\
0 & 0 & 5 & 0 & 0 \\
3 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 &... | Replace those entries by unknowns, say $a,b,c,d$, then calculate the determinant by expanding along the first column. See what you get.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Extension of natural transformation from a dense subcategory I'm trying to prove that the free cocompletion of a small category $\mathcal{C}$ gives an equivalence of categories
$$Cat_+[\widehat{\mathcal{C}},\mathcal{D}] \longrightarrow Cat[\mathcal{C},\mathcal{D}]$$
by precomposition with the Yoneda embedding $h$ (take... | A teacher of mine gave me this answer.
Define for each presheaf $P$, the morphism $\alpha_P : FP \rightarrow GP$ in the unique possible way to make the naturality square of $\alpha$ commutative for all $\lambda_{C,p}$ (these $\lambda_{C,p} : [-,C] \rightarrow P$ form the colimiting cone that establishes $P$ as a colimi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1736967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Puzzle About Cubes (from the book thinking mathematically) I want to confirm my solution to the given problem (solutions were not available in the book)
I have eight cubes. Two of them are painted red, two white, two blue
and two yellow, but otherwise they are indistinguishable. I wish to
assemble them into one la... | The requirement is to have each color on each face of the composed cube.
What that means is that every pair of cubes of the same color must be arranged "diagonally" that is, touching corner to corner, or, in other words, if I put a red cube on the front-bottom-left then the other red cube must be placed on the rear-top... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
Evaluating Line Integral with Green's Theorem I'm given a line integral
$$\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy$$
where C is the circle $$x^2+y^2=8$$ oriented in the counter clockwise direction. I'm supposed to solve it with Green's Theorem.
What ... | Green's theorem converts your line integral into a double integral over the region bounded by the (closed) curve, in your case the circle.
$$\oint_{C^+} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dx\, dy$$
Calculating the line integral itself is hard, but ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Combinatorial interpretation of the sum $\sum s1(n, i+j) \cdot {i + j \choose i} $ I'm trying to figure out a combinatorial interpretation of the following sum:
$\sum\limits_{i,j} s1(n, i+j) \cdot {i + j \choose i} $
and then a compact formula. The $ s1 $ function denotes the Stirling numbers of the first kind (i.e. nu... | \begin{align}
\sum_{i,j}s_1(n,i+j)\binom{i+j}i
&=
\sum_ks_1(n,k)\sum_{i=0}^k\binom ki=\sum_ks_1(n,k)2^k=(-1)^n(-2)_n=(n+1)!\;,
\end{align}
where $(x)_n$ is the falling factorial $x(x-1)\cdots(x-n+1)$.
So the number of subsets of cycles taken from permutations of $n$ elements is the number of permutations of $n+1$ eleme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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An inequality involving $\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$
$$\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$$
Let $(x, y, z)$ be non-negative real numbers such that $x^2+y^2+z^2=2(xy+yz+zx)$.
Question: Find the maximum value of the expression above.
My attempt:
Since $(x,y,z)$ can be non-negative, we can take $x=0$... | Let $P$ be the expression we want to maximise.
Using the following notation: $S_1=x+y+z$, $S_2=xy+xz+yz$ and $S_3=xyz$.
From the hypothesis we get that, $S_1^2=4S2$.
So the expression we want to maximise is:
$P=\dfrac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}=\dfrac{S_1^3-3S_1S_2+3S3}{2S_1S_2}$
Then, simplify it using the h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Problem applying Simpson's rule I am having a problem applying composite Simpson's rule for the integral $$I=\int_0^2\dfrac{1}{x+4}dx$$ with $n=4$.
The exact value of the integral is about $0.405$, however, Simpson's is giving $0.8$, and by increasing the number $n$ up to $8$, Simpson's gives $1.6$ !!
The formula I'm u... | The factor should be
$$
\frac{(b-a)}{3n}
$$
with $b-a=2$ and $n=4$ you get $1/6$ and not $1/3$. One half of $0.8$ is $0.4$ which is close to the exact value.
You can quickly check the correctness of the factor by using the constant function $f=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Number of pairs of rational numbers that satisfy the given relation The number of pairs $(x,y)$ that satisfy : $2x^2 + y^2 + 2xy - 2y + 2 = 0$ is
a.) $0$
b.) $1$
c.) $2$
d.) None of the foregoing numbers
My attempt : I am not well versed in number theory , thus I took the most basic approach that I could see , that is ... | $2x^2 + y^2 + 2xy - 2y + 2 = 0$
$2(x+\frac{y}{2})^2+\frac12(y-2)^2=0$
so the answer is unique, and
$y=2,x=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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All possible ways to order numbers in an array with decreasing rows and columns Given positive integer numbers $1,2,...,N\cdot M$. How many ways are there to order them in an $N\times M$ array given that the values decrease in each row from left to right and in each column from top to bottom? For small arrays one can j... | This is the number of standard Young tableaux for a Young diagram with $N$ rows and $M$ columns. By the hook length formula, this is
$$
\frac{(NM)!}{\prod_{i=1}^M\prod_{j=1}^N(i+j-1)}\;.
$$
This is OEIS sequence A060854. That entry gives the alternative formula
$$
(NM)!\prod_{k=0}^{N-1}\frac{k!}{(M+k)!}\;.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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How to simplify the expression $(\log_9 2 + \log_9 4)\log_2 (3)$ Our test asked to simplify $(\log_9 2 + \log_9 4)\log_2 (3)$.
I simplified the first parenthesis to be $\log_9 (8)$.
So, now I have $\log_9 (8) \cdot \log_2 (3)$ and I can change to base $10$ and get, $$\frac{\log 8}{\log 9} \cdot \frac{\log 3}{\log 2}$$ ... | $$(\log_{9}2+\log_9{4})\log_{2}3$$
this equal to:
$$\log_{9}8\cdot \log_{2}3=\frac{3\log2}{2\log3}\cdot\frac{\log3}{\log2}=\frac{3}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1737990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Potential Frobenius automorphism question Let $F$ be a finite field of characteristic $p$ of size $p^n$
for $n \ge 1$ with the base field $K \cong Z_p$.
I'm attempting to prove that the map $\phi: F → F$ sending $u$ to $u^p$ for each $u \in F$ is a $K$-automorphism of $F$ of degree $n$.
The thing is, I'm fairly certain... | You are correct that $\phi$ is the Frobenius automorphism. Of course, you have to show that it actually is an automorphism. If $\mathbb{F}$ is a field of characteristic $p$, the map $\phi : \mathbb{F} \to \mathbb{F}$ is always a field homomorphism, however it is not always an automorphism.
Show that $\phi$ is a field h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is the Klein bottle homeomorphic to the union of two Mobius bands attached along boundary circle? Question: Determine whether the Klein bottle is homeomorphic to the union of two Mobius bands attached along their boundary circles.
The Klein bottle is the quotient space
$$
K=I^2 /{\sim}, \quad (x,0)\sim(x,1), \; (0,y)\... | This is a diagram of the Klein bottle, note that the diagonal lines divide it into 2 Möbius strips sharing a boundary:
So the answer is yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Upper half-plane $\overline{\mathbb{H}}$ with two boundary punctures Consider $\overline{\mathbb H}$ with two puncture $P_1$ and $P_2$ on the real line, with coordinates $z = x_1$ and $z = x_2$, respectively. Consider another copy of $\overline{\mathbb H}$ with two punctures $P_1$ and $P_2$ on the real line, with coord... | It's enough to consider the case $x_1^{\prime}=0, x_2^{\prime}=\infty$. If $x_1>x_2\in \mathbb{R}$, the map
$$ z\mapsto \frac{z-x_1}{z-x_2} $$
sends $x_1$ to $0$ and $x_2$ to $\infty$, and is conformal because the determinant of the corresponding matrix is $x_1-x_2>0$. And since
$$ \Im\frac{z-x_1}{z-x_2}=\frac{(x_1-x_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Probability question using no-memory property of exponential distribution
A customer must be served first by server 1, then by server 2, and
finally by server 3. The amount of time required for service by server
i is an exponential random variable with rate µi , for i = 1, 2, 3.
Suppose you enter the system when... | I'm not sure if this is correct, but you can do this with a conditioning argument: $E(T_1+T_2+T_3)+E(S|S>T_1+T_2)P(S>T_1+T_2)+E(S|S<T_1+T_2)P(S<T_1+T_2)=\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3}+\frac{\mu_1}{\mu_1+\mu_3}\frac{\mu_2}{\mu_2+\mu_3}\frac{1}{\mu_3}$, since $E(S|S<T_1+T_2)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Canonical Projection I hope are well. I have some doubts as I am new in algebra.
Let $V/W$ the quotient vector space with the usual sum and product, in addition to the equivalence relation defined on $W$ which is a subspace of $V$.
How I can prove that the canonical projection that sends a vector $v$ of $V$ to its equ... | I recommend, investigate the universal property of quotient vector space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Some commutator identities On the way to study Lang's algebra, I cannot solve this problem. See page 69.
Let G be a group and denote the commutator of x and y by & $[x,y]=xyx^{-1}y^{-1}$.
I wanna prove that if $[x,y]=y, [y,z]=z, [z,x]=x$ then $x=y=z=e$.
I tried before posting it, but I don't have a clue. Please give... | Here is the approach I outlined in the comments above.
From the identity $[x,y]=y$, one sees that $xyx^{-1} = y^2$. I will write $y^x$ from now on, for $xyx^{-1}$; similarly, I write ${}^xy$ for $x^{-1}yx$.
We also see from this commutator relation that ${}^yx=yx$ and $x^y=y^{-1}x$. Similar identities can be deduced f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expanding $E[N^2]-E[N]$ I'm trying to prove $$\sum_{i=0}^{\infty}iP[N>i]=\frac{1}{2}(E[N^2]-E[N])$$
by expanding both the RHS and LHS and showing that they are equal. The first thing I did was multiply both sides by $2$ to get $$2\sum_{i=0}^{\infty}iP[N>i]=E[N^2]-E[N]$$
which made it simpler to expand the LHS. I ended ... | Yes, what you wrote $\Bbb E[N^2]-\Bbb E[N]=\Bbb E[N(N-1)]$ is correct. You can expand it as $$\Bbb E\left[N(N-1)\right]=\sum_{i=1}^{\infty}i(i-1)P(N=i)=2P(N=2)+6P(N=3)+12P(N=4)+\dots$$ Now, go back and check where this $14P(X=4)$ comes from in the expansion of the LHS. It should be $12P(X=4)$.
Your expansion of $\Bbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$ then $\lim_{n\to\infty}a_n=?$ I have an objective type question:-
If $$a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$$ then $\lim_{n\to\infty}a_n=?$:-
A)$0$
B)limit does not exist
C)$\frac{1}{\sqrt\pi}$
... | $\displaystyle \ln a_n = \sum_{i=2}^{n+1} \ln(1-\frac{1}{\sqrt{i}})$ and $\displaystyle \ln(1-\frac{1}{\sqrt{i}})\sim -\frac{1}{\sqrt{i}}$
Thefore $\sum_{i=2}^{n+1} \ln(1-\frac{1}{\sqrt{i}})$ diverges to $-\infty$
Hence $\ln a_n\to -\infty$
Hence $a_n\to 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1738921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\varphi(m)+ \tau(m)\leqslant m+1$
Prove that $\varphi(m)+ \tau(m)\leqslant m+1$ where $m\in \mathbb N$
I wrote $m:=p_1^{\alpha_ 1}....p_s^{\alpha _s}$
$$\varphi(m)=p_1^{\alpha_1}(p_1-1)...p_s^{\alpha_s}(p_s-1)$$
$$\tau(m)=(\alpha_1+1)...(\alpha_s+1)$$
Then $\varphi(m)+ \tau(m)=\prod\limits_{i=1}^{s}p_i^{\... | Fix $m\in\mathbb{N}$ and let $A=\{a\in\mathbb{N}:a\leq m,\gcd{(a,m)}=1\},$ $B=\{a\in\mathbb{N}:a\mid m\}$. Now $\lvert A \rvert = \varphi(m)$ and $\lvert B \rvert = \tau(m)$. Hence $$\varphi(m)+\tau(m) = \lvert A \rvert + \lvert B \rvert = \lvert A \cup B \rvert + \lvert A\cap B \rvert.$$
But now note that $A\cup B\sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Finding the Dimension of a given space $V$ I am unsure how to solve this problem:
If $\vec{v}$ is any nonzero vector in $\mathbb{R}^2$, what is the
dimension of the space $V$ of all $2 \times 2$ matrices for which
$\vec{v}$ is an eigenvector?
What I have so far is:
$$ \left[
\begin{array}{ c c }
a & b \\
... | Another solution (using the fact that the vector is non-zero, and we are in dimension 2) is to express the fact that
$ \left[\begin{array}{ c c } a & b \\ c & d \end{array} \right]
\left[\begin{array}{ c } v_1 \\ v_2 \end{array} \right]$
and $\left[\begin{array}{ c } v_1 \\ v_2 \end{array} \right]$
are colinear by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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A question about Rudin's proof of Lusin's theorem In page 56 of Rudin's Real and Complex Analysis, it's stated:
[I]f $f$ is a complex measurable function and $B_n=\{x:|f(x)|>n\}$, then $\bigcap B_n= \varnothing$.
My question is why?
| By definition, a complex function is a map $X\to\Bbb C$. If $f$ is complex, then the set $\bigcap B_n$ is the set of all points $x$ such that $|f(x)| > n$ for every $n\in\Bbb N$. There is no complex number $z$ with the property that $|z|>n$ for every natural number $n$. Hence there is no such $x\in\bigcap B_n$, so $\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Asymptotic estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$ How to find an estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$.
Since $2^{\omega(n)}$ is multiplicative, computing its value at prime power, we see that $2^{\omega(n)}=\sum_{d\mid n}\mu^... | It is not difficult to see that $$\sum_{d\leq x}\mu^{2}\left(d\right)=x\frac{6}{\pi^{2}}+O\left(\sqrt{x}\right)\tag{1}$$ (for a proof see here) so by Abel's summation we have $$\sum_{d\leq x}\frac{\mu^{2}\left(d\right)}{d}=\frac{\sum_{d\leq x}\mu^{2}\left(d\right)}{x}+\int_{1}^{x}\frac{\sum_{d\leq t}\mu^{2}\left(d\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Why does this way of solving inequalities work? Here is what I had to prove.
Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$.
Here is how my teacher did it:
First assume that it is in fact, true that $a^2+b^2 \geq 2ab$. Therefore $a^2+b^2-2ab \geq 0$ . We have $(a-b)^2$ is greater than or equal... | You can not actually. You should rather start with $(a-b)^2 \geq 0$ and the show that $a^2+b^2 \geq 2ab$, not the other way round.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 10,
"answer_id": 1
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Find all pairs of prime numbers $p, q$ such that $p+q = 18(p−q)$. Find all pairs of prime numbers $p, q$ such that $p+q = 18(p−q)$.
It is clear that $p-q$ must be an even number since if we consider $q$ as $2$, we won't get any solution. So any pair of odd prime does the work. I got $p=19$ and $q=17$ as one pair, consi... | $p+q = 18p - 18q$
$17p = 19q$
Therefore $p = 19n$ and $q=17n$.
Then $p$ and $q$ are prime only when $n=1$.
Hence the only solution in primes is $(p,q) = (19,17)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Noether normalization lemma proof I would like to prove the following statement without using Noether normalization lemma (cause it is actually the base case in the induction process of the proof of this lemma).
Let $k$ a field with an infinity of elements, and $A=k[a_1]$ a finitely generated $k$-algebra. Then there e... | Noether normalization says:
Let $k$ be an infinite field, $A = k[a_1, ... , a_n]$ a finitely generated $k$-algebra. Then for some $0 \leq r \leq n$, there exist $r$ elements $b_1, ... , b_r \in A$, algebraically independent over $k$, such that $A$ is finitely generated as a module over $k[b_1, ... , b_r]$.
If $S \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fisherman Combinations This is a real problem. Ten fishermen are going fishing for nine days. Each day, the ten will split into five pairs. For example, on Day 1 Fisherman A will fish with B, C with D, E with F, G with H, I with J. How should the fishermen pair off each day so that each fisherman fishes with every ... | The problem is $n$ fisherman, $n-1$ days.
We need even $n$ for this to make sense, the following construction works: Take a regular polygon with $n-1$ sides. Then pick one color for each of the $n-1$ sides of the polygon, color each diagonal with the color of the side that is parallel to it. Then every vertex will have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f$ given $f$ positive and measurable I'm learning about measure theory (specifically Lebesgue intregation) and need help with the following problem:
Let $f:\mathbb{R}\rightarrow[0,+\infty)$ be measurable and let $\{E_n\}$ be a collection of pairwise disj... | Set $f_N=\sum_{n=1}^{N} f\chi_{E_n}$.
As $f\chi_{E_n}\geq 0$ for each $n$, $f_1\leq f_2 \leq f_3 \leq f_4....$.
Now observe that $f_n \rightarrow f \chi_{E}$ as $n\rightarrow \infty$, where $E=\bigcup_{n=1}^{\infty}E_n$. Thus by MCT we obtain the following equality:
$\lim_{n} \int f_n d\mu= \int f\chi_{E} d\mu$.
Which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1739995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
question on vector calculus notation I just have a question about the vector calculus notation:
$$(u \cdot \nabla)u$$ Is that the same as $( \nabla \cdot u)u$?
| No, these are not the same. The vector $(u\cdot\nabla)u$ is the directional derivative of $u$ in the direction of $u$. It may not be (and probably isn't) parallel to $u$ at each point. The vector $(\nabla \cdot u)u$ is $u$ multiplied by its divergence. It is always parallel to $u$ at each point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How would I find the characteristic equation of this Recurrence Relation? Find and solve a recurrence relation for the number of $n$-digit ternary sequences with no consecutive digits being equal.
Since for ternary, meaning only $3$ possible entries for each space, e.g. $0$, $1$, $2$, the first slot $n$ has $3$ possibl... | In the usual way...just assume that $a_n = A\cdot r^n$ therefore you have:
$$
a_n = 2a_{n-1} \rightarrow Ar^n = 2A\cdot r^{n - 1} = \frac{2A\cdot r^n}{r}\\
1 = \frac{2}{r} \\
r = 2
$$
So $a_n = A\cdot2^n$ (which makes perfect sense since that recursion relation is clearly a simple exponential with common ratio of $2$).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Question about idempotent matrices.
Let $E$ be the $m \times m$ matrix that extracts the "even part" of an $m$-vector $Ex = (x+Fx)/2$, where $F$ is the $m\times m$ matrix that flips $[x_1,\dotsc ,x_m]^{T}$ to $[x_m,\dotsc ,x_1]^T$. Is $E$ idempotent?
| Yes, of course. What happens if you extract the "even part" of a vector which is already "even"?
$$E x=\left[\begin{matrix}(x_1+x_m)/2 \\ (x_2+x_{m-1})/2 \\ \vdots \\ (x_m+x_1)/2 \end{matrix}\right]$$
$$E^2 x=\left[\begin{matrix}((x_1+x_m)/2 + (x_m+x_1)/2)/2 \\ ((x_2+x_{m-1})/2 + (x_{m-1}+x_2)/2)/2 \\ \vdots \\ ((x_m+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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row space and kernel of a matrix A Given a real $m \times n$ matrix $A$ and vectors $x,z \in R^n$ how can I show that
$x \in \ker{A} \wedge x^Tz = 0 \quad \Rightarrow \quad \exists y \in R^m : z = A^Ty $ ?
I thought to start with
$Ax= 0$ and left multiply each side by a vector $ y \in R^m$ to obtain $y^TAx= 0$. The e... | You have to show that $z\in \text{Im}(A^T)$. Recall that $\ker(A)=\text{Im}(A^T)^{\perp}$. Hence $x\in \text{Im}(A^T)^{\perp}$. Since $x^Tz=0$, $\left\langle x,z\right\rangle =0$, so $x\perp z$. Hence $z\in (\text{Im}(A^T)^{\perp})^{\perp}=\text{Im}(A^T)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.
Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.
I can explain the answer but would like help translating i... | Let $A\in\mathbb Z^{n\times n}$ such that $A^{-1}\in\mathbb Z^{n\times n}$. Note that the determinant of an integer matrix is an integer, so $\det\colon\mathbb Z^{n\times n}\to \mathbb Z$. Now, $1=\det(\mathbb I)=\det(A\cdot A^{-1})=\det(A)\cdot\det(A^{-1})$. Since both $\det(A)$ and $\det(A^{-1})$ are integers, they c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Find $\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$ I came across the integral $$\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$$ in a calculus textbook. At this point in the book, only u-substitutions were covered, which brings me to think that there is a clever substitution that one can use to knock off this integral.
I was ab... | One option is to substitute
$$x=2\sin(u)$$
$$dx=2\cos(u)du $$
This means our integral is now
$$=-2 \int _{-\pi/6}^{\pi/6} \frac{2 \cos(u) (\cos^2(u)^{1/2})}{2\sin(u)+3} $$
If we simplify
$$4 \int _{-\pi/6}^{\pi/6} \frac{(\cos^2(u))}{2\sin(u)+3} $$
Now we carry out another substitution of
$$s=\tan(u/2)$$
$$ds=1/2 \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics? I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative.
But today I made a few Google searches, and they all say t... | When multiplying value A by a positive value B, the sign of the result is identical to the sign of A:
*
*If A is positive, then the result is positive
*If A is negative, then the result is negative
When multiplying value A by a negative value B, the sign of the result is opposite to the sign of A:
*
*If A is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$? The question is if $f(x)=x^2$ and $g(x)=x\sin x+\cos x$ then number of points where $f(x)=g(x)$?
My approach:-
$$f(x)=g(x)\implies x^2=x\sin x+\cos x\implies x^2-x\sin x-\cos x=0$$
Let $$h(x)=x^2-x\sin x-\cos x$$Now we have to find out roo... | Equality occurs at the roots of
$$h(x):=f(x)-g(x)=x^2-x\sin(x)-\cos(x).$$
For a continuous function, the minima and maxima alternate and there is at most one root in-between, one before the first and one after the last. A root is there if the function changes sign in the corresponding interval.
Then
$$h'(x)=2x-x\cos(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proof of $[(a \; \text{mod} \; n)+(b \; \text{mod} \; n)] \equiv (a+b)\; \text{mod}\; n$ I'm currently self-studying a course in cryptography, and i understand the importance of understanding modular arithmetic fully. I have proved many operations on modular arithmetic, but one i am stuck on is why:
$[(a \; \text{mod} ... | Let
$$a=q_an+r_a$$
$$b=q_bn+r_b$$
for quotients $q_a,q_b$ and remainders $0\le r_a,r_b<n$ of $a,b$ modulo $n$.
Then
$$\begin{align}
a+b&=(q_a+q_b)n+(r_a+r_b)\\
&=\left(q_a+q_b+\delta\right)n
+\left(r_a+r_b-\delta n\right)
\end{align}$$
for $$\delta=\left\lfloor\frac{r_a+r_b}{n}\right\rfloor$$ where
$\lfloor x\rfloor$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Inverse Vectorization Vec^-1 Hope that you will find this post in good health.
I am Mr. Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using.
During solving that problem, i am facing one problem to calculate inverse vectorization/stacking oper... | I do not really understand what your problem is?
Let $A$ be defined as:
$$A = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$$
Then $\operatorname{vec}(A)$ is defined as
$$\operatorname{vec}(A) = \begin{bmatrix} a_{11} \\ a_{21}\\ a_{12} \\ a_{22} \end{bmatrix}$$
As a result the inverse is defined as
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is a clever way to show that if $0 \leq x \leq 1$ then $x^n \leq x$ for every $n > 1$ ? ($n \in \mathbb{R}$) What is a clever way to show that if $0 \leq x \leq 1$ then $x^n \leq x$ for every $n > 1$ ? ($n \in \mathbb{R}$)
I tried to do it with derivatives but I didn't manage to show why this is true...
| Use induction on $n$:
*
*$n=1$: ok
*$n \to n+1$: By induction, $x^n \le x$. Multiply by $x \le 1$ and get $x^{n+1} \le x$. Note that $x\ge0$ is essential here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
the appropriateness of t-test Two different Universities record data on students who are unable to attend classes due to illness. University 1 recorded absences over ten consecutive days. This data is recorded as N1 below. University 2 recorded absences over six consecutive days. This data is recorded as N2 in below.
... | b) The t-test (or z-test for that matter) are generally only used if the samples are coming from a population that is normally distributed, or if the samples are relatively large. In this case, there is no indication from the question that the number of absences per day follows a normal distribution, and the sample si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to prove that the g.c.d is equal to the prime factorization raised to the minimum of two powers for the prime factorization of $a$ and $b$ as $$a = p_1^{\alpha_1}p_2^{\alpha_2}\cdots{p_t}^{\alpha_t}$$ and $$b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_s^{\beta_s}.$$ I want to prove that $d = gcd(a,b)$ is equal to $${p_... | Remember what divisibility means: $d$ divides $a$ if I can find some integer $c$ such that $dc = a$.
Let's look at one term of your $\frac{a}{d}$ fraction: $\frac{p_1^{\alpha_1}}{p_1^{min(\alpha_1, \beta_1)}}$.
Since $min(\alpha_1, \beta_1) \le \alpha_1$, this fraction has an integer value of $p_1^{\alpha_1 - min(\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Given three coordinates (a,b,c), (d,e,f), and (l,m,n), what is the center of the circle in the 3D plane (h,k,i) that contains these three points. I have tried the following:
$$(a-h)^2+(b-k)^2+(c-i)^2=r^2$$
$$(d-h)^2+(e-k)^2+(f-i)^2=r^2$$
$$(l-h)^2+(m-k)^2+(n-i)^2=r^2$$
Subtracted equation 2 from 1, equation 3 from equa... | For points $(a,b,c)$ and $(d,e,f)$, their perpendicular bisector can be found by:
$$\begin{align*}
(x-a)^2 + (y-b)^2+(z-c)^2 &= (x-d)^2 + (y-e)^2 + (z - f)^2\\
a^2-2ax+b^2-2by+c^2 -2cz &= d^2 - 2dx + e^2 - 2ey + f^2 - 2fz\\
2(a-d)x +2(b-e)y + 2(c-f)z &= a^2+b^2+c^2-d^2-e^2-f^2
\end{align*}$$
Do the same and find the pe... | {
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"url": "https://math.stackexchange.com/questions/1741496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
} |
Why do division algebras always have a number of dimensions which is a power of $2$? Why do number systems always have a number of dimensions which is a power of $2$?
*
*Real numbers: $2^0 = 1$ dimension.
*Complex numbers: $2^1 = 2$ dimensions.
*Quaternions: $2^2 = 4$ dimensions.
*Octonions: $2^3 = 8$ dimensions.... | The particular family of algebras you are talking about has dimension over $\Bbb R$ a power of $2$ by construction: the Cayley-Dickson construction to be precise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Joint distribution of multivariate normal distribution So the question asks: Let $X = (X_1, ... ,X_{2n})$~ $ N (0, ∑)$ (multivariate normal distribution with mean vector $(0,..., 0)$ and covariance matrix $∑$ ), where $n≥ 1$. Find the joint distribution of the vector $(X_1 +... + X_n,X_{n+1} + ... + X_{2n})~ $.
So... | Write
\begin{align}
\begin{pmatrix}
Y_1 \\
Y_2
\end{pmatrix} :=
\begin{pmatrix}
X_1 + \cdots + X_n \\
X_{n + 1} + \cdots + X_{2n}
\end{pmatrix} =
\begin{pmatrix}
1 & \ldots & 1 & 0 & \ldots & 0 \\
0 & \ldots & 0 & 1 & \ldots & 1
\end{pmatrix}X := AX.
\end{align}
Then use if $X \sim N(\mu, \Sigma)$, then $AX \sim N(A\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve $\cos 2x - 3\sin x - 1 = 0$ using addition formula
Solve $\cos 2x - 3\sin x - 1 = 0, \quad 0^{\circ} \le x \le 360^{\circ}$
\begin{align} \cos 2x - 3\sin x - 1 = 0
&\iff 1 - 2\sin^2 x - 3\sin x - 1 = 0 \\
&\iff- 2\sin^2 x - 3\sin x = 0 \\
&\iff2\sin^2 x + 3\sin x =0\\
&\iff\sin x(2\sin x + 3) =0 \\
&\iff\s... | $\sin x =0 \Leftrightarrow x =\pi n (0^{\circ}, 180^{\circ}, 360^{\circ})$
$2\sin x=-3 \Rightarrow \sin x = -\frac 32 - $impossible
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Basis & Dimension for Joint Subspaces Assumption: Assume that $V_1$ and $V_2$ are subspaces of $\mathbb{R}^\mathbb{3}$
Question: "Suppose that $V_1$ is the subspace of $\mathbb{R}^\mathbb{3}$ given by
$V_1 = \{(2t-s, 3t, t+2s): t, s \in \mathbb{R}\}$ and
$V_2$ is the subspace of $\mathbb{R}^\mathbb{3}$ given by ... | The basis for $V_1$ and $V_2$ should be $\{(2,3,1), (-1,0,2)\}, \{(1,0,0), (0,1,1)\}$, respectively. These are two dimensional planes. Their intersection in general should be a $1$-dimensional line.
To find the intersection, you can transform them to equations in terms of $x,y,z$. For $V_1$, we have $x=2t-s, y=3t, z=t+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1741950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
subsets of non regular language I know that there are many languages that are context free but not regular
like $\{a^n b^n :n>0\}.$
But I want to know if every context free but non-regular language has infinitely many non-regular subsets?
Thank you.
| Yes.
If $L$ is context free but non-regular, then for every positive integer n, let $L_n\subseteq L$ be a language such that $|L-L_n|=n$. If $L_n$ were regular, then $L$ would be regular, since an automaton which decides $L_n$ could be (non-deterministically for simplicity) expanded to decide $L_n$ and exactly all the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A basic inequality: $a-b\leq |a|+|b|$ Do we have the following inequality:
$$a-b\leq |a|+|b|$$
I have considered $4$ cases:
*
*$a\leq0,b\leq0$
*$a\leq0,b>0$
*$a>0,b\leq0$
*$a>0,b>0$
and see this inequality is true. However I want to make sure about that.
| Use the triangle inequality:
$$
a - b \leq \vert a - b \vert \leq \vert a \vert + \vert b \vert.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Conditional probability with "at least" We split 8 colored (and distinguishable from each other[each ball is unique]) balls to 4 kids, 2 balls for each kid.
There are 2 blue balls, 2 red balls, 2 yellow balls, 2 green balls [still each ball is unique]
A) It is known that Amy got at least 1 red ball, what is the
pr... | We solve the first problem. Let $A$ be the event Amy got at least one red, and $B$ the event John got at least one red. We are asked to find $\Pr(B\mid A)$, which by definition is equal to $\Pr(A\cap B)/\Pr(A)$.
We compute the two required probabilities. You found $\Pr(A)$ correctly. Now we need $\Pr(A\cap B)$. This is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed form solution for generating function The recursion formula for some probability $P_n(s)$ is $$P_{n+1}(s) = qP_n(s+1) + pP_n(s-1).$$ Define the generating function $$G(z,n) = \sum_{s=-\infty}^{\infty} z^s P_n(s)$$ and prove the recursion relation $$G(z,n+1) = (pz + qz^{-1}) G(z,n)$$ Obtain a closed form soluti... | Observe that $G(z,n)$ is a geometric sequence for fixed $z$ hence $G(z,n)=(pz+qz^{-1})^nG(z,0)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why Can we work with $M$ model countable transitive model of some finite fragment of $\mathrm{ZFC}$ and why is it exist.? When we say that let $M$ be a countable transitive model of some finite
fragment of $\mathrm{ZFC}$.
Why Can we work with $M$ model countable transitive model of some finite
fragment of $\mathrm{ZFC... | If $T$ is a finite fragment of $\mathsf{ZFC}$, then by the reflection theorem there are infinitely many ordinal $\alpha$ so that $V_\alpha \models T$. Then by the Downward Loweinheim Skolem theorem, there is a countable elementary substructure $N \prec V_\alpha$ so $N \models T$. Let $M$ be the Mostowski collapse and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Unique solution for circuits in Linear Algebra A standard application of Linear Algebra is circuits and Kirchhoff's Laws. Does anyone know of a proof of uniqueness of a solution of a system given by these laws? There are many, many examples, but little theory regarding why there is always a unique solution.
For referen... | There is a statement that works for your purposes (proposition 9.4) presented in Markov Chains and Mixing Times (Levin, Peres, Wilmer) that you can follow by reading pages 115-118.
Perhaps there are other, better references for you purpose, but this is the first that comes to mind.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding conditions on the eigenvalues of a matrix Consider the $2\times2$ matrix
$$A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$$
where $a,b,c,d\ge 0$. Show that $\lambda_1\ge\max(a,d)>0$ and $\lambda_2\le\min(a,d)$.
So the eigenvalues are given by the characteristic polynomial
$$(a-\lambda)(d-\lambda)-bc=0\implies \lamb... | Note that by the formula you derived one of the conditions implies the other. So suppose $\lambda_1\geq \lambda_2$. Now you know that
$$ \begin{pmatrix}
a-\lambda_1 & b\\
c & d-\lambda 1
\end{pmatrix} $$
is singular, i.e. there is a real number $r$ s.t.
$$ \begin{pmatrix}
a-\lambda_1 \\
c
\end{pmatrix} =r\begin{pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1742811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is there an analogue of Frucht's theorem for sandpile groups? In other words, is it the case that for every abelian group $G$, there exists a graph $H$ such that the sandpile group of $H$ is isomorphic to $G$? If not, is the truth of falsity of this still an open question?
| If $G=\bigoplus_{i=1}^N \mathbb{Z}_{k_i}$, then $G$ is the sandpile group of the (multi)graph on $N$ non-sink vertices where the $i$-th vertex is joined with the sink by $k_i$ edges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Topology/ Metric on possibly unbounded functions I am trying to think of a topology (possibly metric, as I am more used to think about things in metric spaces) on possibly unbounded functions (on $\mathbb{R}$) such that
1) convergence in that defined topology implies pointwise convergence and
2) the limit of continuo... | Here is a metric which describes uniform convergence on $\mathbb R$: $D(f,g)=\sup\lbrace \min\lbrace|f(x)-g(x)|,1\rbrace: x\in\mathbb R\rbrace$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How do I find the equation of an osculating circle when I'm given the parabola? This is a question given out by my calculus professor, and I'm completely stumped as to how I need to go about solving it.
Let the parabola $y=x^2$ be parameterized by $r(t)=ti+t^2j$. Find the equation of the osculating circle for the parab... | As the comment from Rory Daulton said, you can look up the formulas for calculating curvature here. These same formulas must be in your class notes or textbook, too, or else your teacher wouldn't be asking you this question.
There are two choices for the curvature formula: one if you choose to think of the curve in par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate intersection and union of probabilities? lets say I have a switch A with 3 legs, each leg has 0.8 chance to be connected (and then electricity will flow), we need only 1 leg connected for A to transfer the electricity (so sorry I didn't explain it that well I'm having hard time to translate this proble... | Hint One of the three legs running isnt dependent on the running of other two legs. So the events are independent events. So we just multiply their probabilities instead of adding .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How to show that $\lim_{n \to \infty}\frac{2^{n^2}}{n!} = \infty$? I know $\lim \limits _{n \to \infty}\frac{2^{n^2}}{n!} = \infty$, but I need to prove it using the definition of limit, show that there is a $M$ such that all $a_n>M$.
I tried looking at $\frac{a_{n+1}}{a_n}$ and found out that $a_{n+1}>\frac{1}{2}a_n$,... | Let's prove that $2^{n^2}\ge(n+1)!$, for every $n$. The case $n=0$ is obviously true. We also have
$$
2^{(n+1)^2}=2^{n^2}\cdot 2^{2n+1}\ge 2^{2n+1}(n+1)!
$$
and we're done if we show that $2^{2n+1}\ge n+2$. Again, the base step is trivial; moreover
$$
2^{2n+3}=4\cdot 2^{2n+1}\ge 4(n+2)=n+3n+8\ge n+3
$$
Therefore
$$
\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Let $\{ y_k \}$ that satisfies $ y_k\le {2^k\over M}y_{k-1}^\beta$ , then $\lim_{k\to \infty}y_k=0$. Let be a sequence $\{ y_k \}^\infty _{k=0} \subset (0,\infty) $ that satisfies
$$ y_k\le {2^k\over M}y_{k-1}^\beta , $$
where $k=1,2,...$, and $\beta\gt 1$ , $M\gt0$.
Prove that if $M\gt2^{\beta\over \beta-1... | Try to compare with
$$
ca^ky_k=u_k\le u_{k-1}^β= (ca^{k-1}y_{k-1})^β
\\\iff\\
y_k\le c^{β-1}a^{(k-1)β-k}y_{k-1}^β=(c^{β-1}a^{-β})(a^{β-1})^ky_{k-1}^β
$$
which is successful by identifying the relations $2=a^{β-1}$, $M^{-1}= a^{-β}c^{β-1}$, that is
$$
a=2^{\frac1{β-1}},\quad c=M^{-\frac1{β-1}}2^{\frac{β}{(β-1)^2}}
$$
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1743905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Making groups of 2., probability of getting a certain group? Let us say we have $n$ people $p_1, ...., p_n$ where $n$ is even. We find some random way to make groups of 2, and we are interested in if $p_i$ gets in a group with $p_j$, if $p_i$ gets $p_k$, $p_j$ gets in a group with $p_k$, and etc. Let us say that we hav... | There are $n-1$ equally likely ways to choose $p_2$'s partner. Of these, $2$ are "favourable."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Two multi-variable limit problems $$\lim_{(x,y)\to(0,0)}\frac{2x^2y}{x^4+3y^2}$$
I'm getting that the limit DNE because using $(0,y)\to(0,0)$ it is $0$ but for $(x,x^2)\to(0,0)$ it is $1/2$. Since $0$ does not equal $1/2$ the limit does not exist.
$$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{2x^2+y^2}$$
I'm getting that the lim... | Yeah, you're right, from showing that you can say that the limit does not exists, from more you can say this in terms of the formal definition of a limit, but, with this approach it's enought to say that the limit does not exists
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problems with integrating a (step) probability density function I've been sitting a embarrassing amount of time over this problem:
I am given a probability density function f(x) like this:
y= 1/6 when x between [-2,-1]
y= 2/6 when x between [-1,1]
y= 1/6 when x between [1,2]
y= 0 else
My task is to find out how prob... | Your answer seems to be correct as $\int_{0.5}^1 \frac{2}{6} dx = 0.5 * \frac{2}{6} = \frac{1}{6}$ and $\int_1^{1.5} \frac{1}{6} dx = 0.5 \frac{1}{6} = \frac{1}{12}$
And the sum of the two is obviously $\frac{3}{12} = \frac{1}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear functional is continuous $\implies$ it is bounded Let $f:X \rightarrow \mathbb R$ be a continuous linear functional. Prove $f$ is bounded.
Since $f$ is continuous, $\forall \varepsilon >0$, there exists $\delta >0$ such that $|f(x)-f(y)|=|f(x-y)|=|f(z)|< \varepsilon$ whenever $|x-y|<\delta$. We let $z=x-y$.
Can... | Since $f$ is continuous (at $0$), there is a neighbourhood $U$ of $0$ such that $f(U)\subset(-1,1)$. Choose $\delta>0$ such that $\{x\in X|\|x\|\leq\delta\}\subseteq U$. Then, if $x\in X$ is such that $\|x\|\leq \delta$, we have $x\in U$, and hence, $|f(x)|\leq 1$. Since $\|\frac{\delta x}{\|x\|}\|=\delta$, it follows ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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