Recent Questions - Mathematics Stack Exchange

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• Is $(ab)^.5 = a^.5*b^.5$ true for all complex numbers?
• Use of kernrel and range.
• The action of the longest element on weights
• How to flesh out a sketch of a proof of a corollary of the Maximum Principle?
• Given $P(\sqrt2)$ = $13+9\sqrt2$, then find $\frac{P(1)}{4}$.
• A problem about Prime Numbers and Perfect Squares
• 'elementary' proof that all matrix with zero trace has the form $AB-BA$
• Trying to understand construction of $C^*-$algebra from a paper
• Why are vector coordinates different to matrix coordinates
• A more efficient way to determine two arcs overlap?
• Constructing new spherical basis functions
• Erdos-Renyi random graphs proof
• limit of Gamma functions using Mathematica
• What are the odds that after hitting shuffle twice on a 155 unique songs playlist on spotify, 3 of the 6 first songs are the same on both shuffle.
• 2nd Order ODE w/ Periodic BCs
• How many numbers from $0$ to $999$ are multiples of $13$ and also have digits that add up to $13$?
• If $C=\sum_{k=0}^\infty\frac1{k!}r^k\cos(kt)$ and $S=\sum_{k=1}^\infty\frac1{k!}r^k\sin(kt)$, show $C\frac{dC}{dr}+S\frac{dS}{dr}=(C^2+S^2)\cos t$
• Evaluating $\sum_{k=-m}^{m}(-1)^k \, e^{-\frac{i\pi k}{m}}\frac{2\left(ze^{\frac{i\pi k}{m}}\right)^2}{\left(ze^{\frac{i\pi k}{m}}\right)^2 - q^2}$
• inequality and series
• Strange Number Base System's Name
• Show that $T : [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] \to [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] $ is mixing.
• Finite Galois group of Galois extension implies that the extension is finite?
• A tough integral $\int_0^{\infty}\frac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $
• $\prod_{i=1}^{n} (3 +\frac{1}{a_i})$ cannot be a power of 2 if $a_i \equiv \pm 1 \pmod 6$
• How does folding paper affect the geometry angles?
• Finding the orthogonal complement of a span?
• Derivative of $\mathrm{L}_2$ norm of function
• Does there always exist a field automorphism sending one root to another? Galois Theory
• Intuition behind this Olympiad problem using generating functions
• Testing whether the cost-cutting measures seem to be working at the $5\%$ significance level

Is $(ab)^.5 = a^.5*b^.5$ true for all complex numbers? Posted: 20 Jun 2021 08:29 PM PDT I have just started learning complex numbers and am now confused about one property. We know that $(ab)^.5=a^.5*b^.5$ is only valid if both $a$ and $b$ are not negative simultaneously. My teacher told me that the relationship breaks if both the numbers are negative to solve the following contradiction. $(-1)^.5*(-1)^.5=((-1)(-1))^.5=1$ Now, that makes me wonder if the relationship also holds if $a$ or $b$ are complex numbers with non-zero imaginary parts. Any help would be greatly appreciated.

Use of kernrel and range. Posted: 20 Jun 2021 08:22 PM PDT Is there any use of the kernel and the range? Any 'aplication' works for me, is it used to prove an important theorem?

The action of the longest element on weights Posted: 20 Jun 2021 08:21 PM PDT I am doing some computations related to some work I do in Lie theory and I need to compute the result of the action $$w_0 \omega_{i},$$ where $w_0$ denotes the longest element in the Weyl group $W$ of a semisimple simply-connected complex algebraic group $G$ and the $\omega_i$'s denote the fundamental weights. I know I can express $w_0$ as a product of reflections $s_j$ and consider the action of each $s_j$ one by one, but this takes me a quite a lot of time and effort. Is there a shortcut (or a program) to compute this?

How to flesh out a sketch of a proof of a corollary of the Maximum Principle? Posted: 20 Jun 2021 08:26 PM PDT Today I have been looking at the Maximum Modulus Principle (MMP). I take it for granted in the following form. $\quad(*)\quad$ Let $f$ be holomorphic on a domain $\Omega \subset \mathbb{C},$ where "domain" indicates open and connected. Let $\zeta \in \Omega,$ and suppose it has a neighbourhood in which $|f(\zeta)| \geq |f(z)|$ holds. Then $f$ is constant on $\Omega.$ On the wikipedia page for MMP, as of today the 20th of June, there is a corollary of $(*)$ given, and a sketch of the proof. The corollary is called there the "Related statement" (link to it: https://en.wikipedia.org/w/index.php?title=Maximum_modulus_principle&oldid=1009743623#Related_statement ). I tried to write a more detailed proof of how the corollary follows from $(*)$, because I wasn't fully convinced by what I saw there. My question is whether my proof (below) is actually what the sketch is suggesting. If "yes", then maybe you can verify that my proof is alright. But if "No", what is going on with that sketch? Is there some lemma being implicitly used in the reasoning? I'd like to know what I'm missing. Proof: Let $D$ be bounded, non-empty, open, so that $\overline{D}$ is compact. We are given that $f$ is continuous on $\overline{D}$ and is holomorphic on $D$. Since $|f|$ is real and continuous on a compact, there is a $z_o \in \overline{D}$ where the modulus attains its maximum. If $z_o \in \partial D$, then we are done; else we have $z_o \in D$. Let $D_m$ be the connected component of $D$ in which $z_o$ lies. (We take it as known that $D$ may be partitioned into a family of disjoint domains.) By $(*)$ we have that $f$ is constant on $D_m$, and it is constant on $\overline{D_m} \subset \overline{D}$, by continuity. Thus $|f|$ attains its maximum at some $x \in \partial D_m.$ ($\mathbb{C}$ is connected, $D_m$ is bounded; hence $\partial D_m \not = \emptyset.$) Lastly we must show that $x \in \partial D.$ Clearly $x$ is not exterior to $D$. So suppose $x \in D_0\not=D_m$, one of the connected components of $D$. Then $x \in \overline{D_m} \cap D_0$, a contradiction. (For $D_0, D_m \subset \mathbb{C},$ open and disjoint, this cannot be.)

Given $P(\sqrt2)$ = $13+9\sqrt2$, then find $\frac{P(1)}{4}$. Posted: 20 Jun 2021 08:26 PM PDT My Question is- Let P($\sqrt2$) = a$_0$ + a$_1$x + ... a$_n$x$^n$ be a polynomial with integer coefficients and a$_i$ $\in$ {$0$,$1$} for all $i$ = $0,1,2,....n$ Given $P(\sqrt2)$ = $13+9\sqrt2$, then find $\frac{P(1)}{4}$. I can't understand what to do with this question. Please help me. Any help will be appreciated.

A problem about Prime Numbers and Perfect Squares Posted: 20 Jun 2021 08:10 PM PDT Can we find all $n$ such that there exists a prime number $p$ s.t. $1+p+p^2+\cdots+p^n$ is a perfect square, where $n$ is a natural number? For $n=1$, when $p=3$, $1+p=4$, which fits our standards. For $n=2$, we can know that $p^2<1+p+p^2<1+2p+p^2=(1+p)^2$, so $1+p+p^2$ cannot be a perfect square. For $n\geqslant3$, I want to use the identity $x^n-1=(x-1)(x^{n-1}+x^{n-2}+\cdots+1)$. If there exists a prime number $p$ s.t. $1+p+p^2+\cdots+p^n\;\;(n\geqslant3)$ is a perfect square, let $1+p+p^2+\cdots+p^n=m^2$. Then, $m^2(p-1)=p^{n+1}-1$. For the $n\geqslant3$ case, I don't what can I do next, or should I split it into more cases. There are several links below that focus on particular cases of $n$. $n=4$ link, another $n=4$ link. Note: $\mathbb{N}=\mathbb{Z}^+$. I want to clarify that I couldn't find any other questions that are similar to my problem. I hope this is not another duplicate.

'elementary' proof that all matrix with zero trace has the form $AB-BA$ Posted: 20 Jun 2021 08:18 PM PDT How to prove the fact mentioned in the title without using the theory of lie algebras? For the record, though questions about this fact has indeed appeared in MSE several times, yet up til now, I think no one has given any 'elementary'(only use linear algebra) proof of this fact on MSE. So it's not a duplicate. Any reference or answer would be greatly appreciated. Thanks in advance.

Trying to understand construction of $C^*-$algebra from a paper Posted: 20 Jun 2021 08:13 PM PDT Recall that a ternary Banach algebra is a complex associative Banach space $A$, equipped with a ternary product $[.,.,.]:A^3 \to A$ which is linear in outer variables and conjugate linear in middle variable and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$ Ternary rings of operators are obvious examples of ternary Banach algebra. This paper (section $3$) discusses construction of $C^*$-algebra associated to ternary Banach algebra. Can someone please explain me the construction given in the paper? P.S: The copy of the same question on mathoverflow can be found here

Why are vector coordinates different to matrix coordinates Posted: 20 Jun 2021 08:24 PM PDT I'm starting a Linear Algebra course and I'm a bit confused. Say we have a vector $x = \begin{pmatrix} x_1\\ x_2\\ \end{pmatrix}$, and another vector $y = \begin{pmatrix} y_1\\ y_2\\ \end{pmatrix}$ When we have a matrix composed of the two vectors, instead of saying $\begin{pmatrix} x_1&y_1\\ x_2&y_2\\ \end{pmatrix}$ we say that the first column corresponds to $x_1$, and the second corresponds to $x_2$ so $\begin{pmatrix} x_1&x_2\\ y_1&y_2\\ \end{pmatrix}$ Maybe I misunderstood something, but I feel like this is a bit confusing to me, may someone explain to me how this works? Thanks in advance!

A more efficient way to determine two arcs overlap? Posted: 20 Jun 2021 07:44 PM PDT I have made myself a simple script that determines if two arcs which have a width overlap: Currently its quite slow and inefficient and was wondering if any one knows a more efficient mathematical solution to solve this? My current setup to solve overlap is done with the following steps: Find intersecting points of the 4 circles (inner and outer radius of both arcs). Check if the intersecting points is inside the angle range of the arcs. Next i do 4 tests to determine if the ends of arcs overlap which is 4 line segments tested against the 4 arcs which represent the inner and outer ranges of the radius of the pieces. Since i do this every frame this is a lot of calculations. I am wondering if any smart person here knows a simpler mathematical approach that might simplify my current solution. My arcs are represented by StartAngle, EndAngle, Origin and MinRadius/MaxRadius. Thanks.

Constructing new spherical basis functions Posted: 20 Jun 2021 07:36 PM PDT I am trying to construct a new set of basis functions on the sphere. The reason is I am working in a special coordinate system $(r,\theta_q,\phi_q)$ where $\theta_q = \theta_q(r,\theta,\phi)$ and $\phi_q = \phi_q(r,\theta,\phi)$ are nonlinear functions of the usual spherical coordinates. The coordinates $(r,\theta_q,\phi_q)$ have a geometry more suitable to my problem. I would like to develop basis functions similar to the spherical harmonics which exhibit the geometry of this new coordinate system. I proceed by defining $$ \alpha_l^m(\mathbf{r}) = Y_l^m(\theta_q(\mathbf{r}),\phi_q(\mathbf{r})) $$ where $Y_l^m$ are the usual spherical harmonics. The $\alpha_l^m$ exhibit the geometrical properties of my coordinate system, however they are not eigenfunctions of the Laplacian like the spherical harmonics. So in the next step, I define: $$ \alpha_l^m(\mathbf{r}) \approx \beta_l^m(\mathbf{r}) = \sum_{k=0}^K \sum_{n=-k}^k d_{lk}^{mn}(r) Y_k^n(\theta,\phi) $$ where I choose a sufficiently high truncation limit $K$, and the coefficients $d_{lk}^{mn}$ are determined from doing a truncated spherical harmonic transform of the $\alpha_l^m$. By choosing $K$ large enough, I can make the error between $\alpha_l^m$ and $\beta_l^m$ as small as I want. Now the last problem is the functions $\beta_l^m$ are functions of radius $r$, so in a final step, I pick some radius $a$ and define my final functions, $$ \gamma_l^m(\theta,\phi) = \sum_{k=0}^k \sum_{n=-k}^k d_{lk}^{mn}(a) Y_k^n(\theta,\phi) $$ There isn't a large variation of the $\alpha_l^m$ with radius, so I can pick $a$ from a wide range of radii and still keep the geometry I want in the basis functions. Now, the functions $\gamma_l^m$ have the nice property that they contain the geometry of my coordinate system, and they are eigenfunctions of the Laplacian as they are just linear combinations of $Y_l^m$. However, it would be even nicer if: 1. The $\gamma_l^m$ obeyed some orthogonality relationship like the spherical harmonics 2. The $\gamma_l^m$ formed a complete set of basis functions I would like to expand other functions in terms of the $\gamma_l^m$, such as: $$ f(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l f_l^m \gamma_l^m(\theta,\phi) $$ and pick a sufficiently large $L$ so that I can get a nice approximation to $f$. If my function $f$ exhibits the same symmetry as my coordinate system $\theta_q,\phi_q$, it seems in practice that I can choose a fairly small $L$ and get a good approximation, though I don't know if that will be true for any function $f$. Also, in order to calculate the $f_l^m$ of such an expansion, I need an orthogonality relationship for the $\gamma_l^m$. However, what I get when doing the usual integration of both sides is: $$ \int f(\theta,\phi) \gamma_k^n(\theta,\phi) d\Omega = \sum_{l=0}^L \sum_{m=-l}^l f_l^m \sum_{r=0}^K \sum_{s=-r}^r d_{kr}^{ns} d_{lr}^{ms} $$ which doesn't seem to lead to a nice way to solve for $f_l^m$. Does anyone know of another way to construct a set of basis functions $\gamma_l^m$, which: Contain the geometry of my coordinate system $(r,\theta_q,\phi_q)$ Are eigenfunctions of the Laplacian in spherical coordinates, Form a complete basis set Have a nice orthogonality relation?

Erdos-Renyi random graphs proof Posted: 20 Jun 2021 07:35 PM PDT I could not find any reference to prove these statements. Any help is appreciated. The average number of edges of G(n,m) is m. The average degree of G(n,m) is 2m/n. The average number of edges of G(n,p) is $n\choose 2$ p. The average degree of G(n,p) is (n − 1)p.

limit of Gamma functions using Mathematica Posted: 20 Jun 2021 08:19 PM PDT I am trying to evaluate: $$ \underset{z\to n}{\text{lim}}\frac{\Gamma (1-z)}{\Gamma (1-2 z)} $$ Here are my steps: \begin{align*} \underset{z\to n}{\text{lim}}\frac{\Gamma (1-z)}{\Gamma (1-2 z)}&=\underset{z\to n}{\text{lim}}\frac{\Gamma (-n-z+1) \prod _{k=0}^{n-1} (-k-z)}{\Gamma (1-2 z)}\\ &=\left(\underset{z\to n}{\text{lim}}\frac{\Gamma (-n-z+1)}{\Gamma (1-2 z)}\right) \underset{z\to n}{\text{lim}}\prod _{k=0}^{n-1} (-k-z)\\ &=(-1)^n \left(\underset{z\to n}{\text{lim}}\frac{\Gamma (-n-z+1)}{\Gamma (1-2 z)}\right) \underset{z\to n}{\text{lim}}\prod _{k=0}^{n-1} (k+z) \end{align*} Here is the problem, also verified by Mathematica, $$ \underset{z\to n}{\text{lim}}\frac{\Gamma (-n-z+1)}{\Gamma (1-2 z)}=1 $$ so we should expect that \begin{align*} (-1)^n \left(\underset{z\to n}{\text{lim}}\frac{\Gamma (-n-z+1)}{\Gamma (1-2 z)}\right) \underset{z\to n}{\text{lim}}\prod _{k=0}^{n-1} (k+z)&=(-1)^n \underset{z\to n}{\text{lim}}\prod _{k=0}^{n-1} (k+z)\\ &=(-1)^nn(n+1)\cdots(2n-1)\\ &=(-1)^n\frac{1\cdot2\cdots n(n+1)\cdots(2n-1)({\color{red} 2}n)}{{\color{red} 2}\cdot 1\cdot2\cdots n}\\ &=\frac{(-1)^n}{2}\frac{(2n)!}{n!} \end{align*} One may, then, conclude that \begin{align*} \underset{z\to n}{\text{lim}}\frac{\Gamma (1-z)}{\Gamma (1-2 z)}=\frac{(-1)^n}{2}\frac{(2n)!}{n!} \end{align*} But Mathematica gives \begin{align*} \underset{z\to n}{\text{lim}}\frac{\Gamma (1-z)}{\Gamma (1-2 z)}={(-1)^n}\frac{(2n)!}{n!} \end{align*} Am I missing something here? Of course, there are other approaches, but I want to understand why! there is such inconsistency.

What are the odds that after hitting shuffle twice on a 155 unique songs playlist on spotify, 3 of the 6 first songs are the same on both shuffle. Posted: 20 Jun 2021 07:35 PM PDT Imagine you have $155$ unique songs in a playlist on Spotify. You hit shuffle. Out of the first $6$ songs, you have $3$ particular songs that we'll name "song A", "song B", and "song C". The order in which these $3$ songs show up within those first $6$ songs doesn't matter. Then, you hit shuffle again and get another set of $6$ songs. $3$ of those $6$ songs are once again A, B, and C. What are the odds of this happening? I'm kinda stuck trying to math this. I know that you have $$\frac{155!}{6!(155-6)!}$$ different ways to select $6$ songs and that would be a denominator for a probability "P". You would then get a numerator and multiply $P$ by itself ($P^2$) because we shuffle $2$ times. Anyways, I think that's what I need to do. The numerator is where I get stuck. Not sure how to calculate it. Anyone can help? Am I on the right track?

2nd Order ODE w/ Periodic BCs Posted: 20 Jun 2021 07:34 PM PDT I am a bit rusty with ODEs and would appreciate some help with the following. I am trying to solve a linear homogeneous PDE with separation of variables. The time component was easy enough. It was an exponential of the form $G(t)=c \exp(-\lambda b t)$ where $c$ and $b$ are positive constants and $\lambda$ is the separation constant (eigenvalue). The PDE is $$ \begin{align} \frac{\partial u}{\partial t} + v\frac{\partial u}{\partial x} &= \nu \frac{\partial^2 x}{\partial x^2}\\ u(0,t) &= u(L,t) \quad u(x,0) = u_o \end{align} $$ Now I want to solve the following 2nd order homogeneous ODE with constant coefficients for the space component $\phi(x)$, with periodic boundary conditions. $$ \frac{d^2\phi}{dx^2} - b \frac{d\phi}{dx} + \lambda \phi= 0 \\ \phi(0) = \phi(L) \quad , \quad \frac{d\phi}{dx}\Bigg|_0 = \frac{d\phi}{dx}\Bigg|_L $$ I know that I want $\lambda$ to be positive so that $G(t)$ decays with time. If I assume a solution for $\phi(x)$ of the form $\exp(rx)$ then I get the characteristic equation $$ r = \frac{b \pm \sqrt{b^2-4\lambda}}{2} $$ I know that I want $\lambda$ to be positive but I don't know whether it is greater than, equal, or less than $b^2$ - I don't know $\lambda$. Okay. I will assume at first that it is greater than $b^2$ which means the characteristic equation has complex roots. In that case I know the solution will have the following form. $$ \begin{align} \alpha &= \frac{-b}{2}\\ \beta &= \frac{\sqrt{4\lambda-b^2}}{2}\\ \phi(x) &= \exp(\alpha x)( C_1 \cos\beta x + C_2\sin\beta x) \end{align} $$ Assuming I'm correct up to this point, this is where I get a little confused. I need to find $C_1$, $C_2$, and $\lambda$. I think I should be able to use the following two equations to determine $C_1$ and $C_2$, right? (two equations, two unknowns) $$ \phi(0) = \phi(L) \quad , \quad \frac{d\phi}{dx}\Bigg|_0 = \frac{d\phi}{dx}\Bigg|_L $$ But I am still left with $\lambda$. Do I have to infer it's form based on the results for $C_1$ and $C_2$? Pick something so that $C_1$ or $C_2$ do not result in the trivial solution?

How many numbers from $0$ to $999$ are multiples of $13$ and also have digits that add up to $13$? Posted: 20 Jun 2021 07:42 PM PDT Can I have a hint on this algebraic question? There are numbers from $0$ to $999$ that are multiples of $13$ and also have digits that add up to $13$. How many numbers are there possible? All I have so far is $100a + 10b + c = ?$ and $a + b + c = 13$. I have no idea what to do next. Please help? Thanks.

If $C=\sum_{k=0}^\infty\frac1{k!}r^k\cos(kt)$ and $S=\sum_{k=1}^\infty\frac1{k!}r^k\sin(kt)$, show $C\frac{dC}{dr}+S\frac{dS}{dr}=(C^2+S^2)\cos t$ Posted: 20 Jun 2021 07:59 PM PDT Please help me with this: If $$C=1+r\cos(\theta)+\frac{r^{2}\cos(2\theta)}{2!}+\frac{r^{3}\cos(3\theta)}{3!}+\cdots \tag1$$ and $$S=r\sin(\theta)+\frac{r^{2}\sin(2\theta)}{2!}+\frac{r^{3}\sin(3\theta)}{3!}+\cdots \tag2$$ then prove that $$C\frac{dC}{dr}+S\frac{dS}{dr}=(C^{2}+S^{2})\cos(\theta) \tag3$$

Evaluating $\sum_{k=-m}^{m}(-1)^k \, e^{-\frac{i\pi k}{m}}\frac{2\left(ze^{\frac{i\pi k}{m}}\right)^2}{\left(ze^{\frac{i\pi k}{m}}\right)^2 - q^2}$ Posted: 20 Jun 2021 08:26 PM PDT From quite some time I'm struggling on proving that: $$\sum_{k=-n}^{n}(-1)^k \, e^{-\frac{i\pi k}{m}}\dfrac{2\left(ze^{\frac{i\pi k}{m}}\right)^2}{\left(ze^{\frac{i\pi k}{m}}\right)^2 - q^2} = 2z^2m\,\dfrac{q^{m-1}z^{m-1}}{z^{2m} - q^{2m}} $$ where $m = 2n+1$. My intuition to prove this would be proving that the partial fraction expansion of the right-hand side coincides with the left-hand side. However I've no idea if that would get us anywhere. I'm looking for a real analytic proof but a proof using complex analysis and residue theorem would be most welcomed. Any help or hints would be highly appreciated. Thanks for reading.

inequality and series Posted: 20 Jun 2021 08:29 PM PDT suppose $x_i>0$ for $i=1,2,\cdots,n$ and $x_1+x_2+\cdots+x_n=1$ show that $$\frac{x_1}{1+x_2+x_3+\cdots+x_n}+\frac{x_2}{1+x_1+x_3+\cdots+x_n}+\frac{x_3}{1+x_1+x_2+\cdots+x_n}+\cdots+\frac{x_n}{1+x_1+x_2+\cdots+x_{n-1}}\ge\frac{n}{2n-1}$$ attempt from the equalitys i could rewrite $1-x_1=x_2+\cdots+x_n$ and then $$\frac{x_1}{1+x_2+x_3+\cdots+x_n}+\frac{x_2}{1+x_1+x_3+\cdots+x_n}+\frac{x_3}{1+x_1+x_2+\cdots+x_n}+\cdots+\frac{x_n}{1+x_1+x_2+\cdots+x_{n-1}}=\frac{x_1}{2-x_1}+\frac{x_2}{2-x_2}+\frac{x_3}{2-x_3}+\cdots+\frac{x_n}{2-x_n}$$ but idk how to proced from there, so how do i solve this? the fact $x_i>0$ makes me assume AM-GM inequality could be used but idk how one would use.

Strange Number Base System's Name Posted: 20 Jun 2021 08:16 PM PDT Consider a base where there are six digits. The first digit denominates the base, and the other five marks the number proper. I was thinking of it working something like this (note that I will put the number in base-10 on the left of the equals sign): 1=200001 2=200011 3=200111 4=201111 5=211111 6=300000 7=300001 8=300002 9=300010 10=300020... Does such a system have a name? I can't imagine it'd be too useful in other areas of maths, but it was something I was thinking about today...

Show that $T : [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] \to [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] $ is mixing. Posted: 20 Jun 2021 07:36 PM PDT Is this shuffling map on the unit interval mixing ? $$ f(x) = \left\{ \begin{array}{ccl} x + (\sqrt{5} + \sqrt{7}) & \text{ if } & 0 < x < \sqrt{3} \\ x + (\sqrt{7} - \sqrt{3}) & \text{ if } & \sqrt{3} < x < \sqrt{3} + \sqrt{5} \\ x - (\sqrt{3}+\sqrt{5}) & \text{ if } & \sqrt{3} + \sqrt{5} < x < \sqrt{3} + \sqrt{5} + \sqrt{7}\\ \end{array} \right. $$ Here $f : [0, \sqrt{3}+\sqrt{5}+ \sqrt{7}] \to [0, \sqrt{3}+\sqrt{5}+ \sqrt{7}] $ yet we could rescale so to be map $f: [0,1] \to [0,1]$. this is a 1-1 map this exchanges the permutations in the interval $\pi = (3,2,1)$. it's called an interval exchange map. Such a map is very likely to be ergodic and weak-mixing. Let's review those definitions. for every measurable set $A \in \mathcal{B}$ we have $T^{-1}(A) \subset A$ implies either $\mu(A) = 0$ or $\mu(A) = 1$. strong mixing $\displaystyle \lim_{n \to \infty} \mu(A \cap T^{-n}B) = \mu(A) \cdot \mu(B) $ so the two sets $A$ and $B$ are eventually "independent". weak-mixing $\displaystyle \lim_{n \to \infty} \frac{1}{n} \left| \sum_{k=0}^{n-1} \mu(A \cap T^{-k}B) - \mu(A)\mu(B) \right|= 0 $ so the "average" over all the over lap with the pre-images should be independent. Maybe one could show that the invariant measure of the interval exchange map, is just Lebesgue measure (which served as a replacement for the Riemann integration). We are guessing as much. Then one could ask what the strong mixing and weak-mixing statements should represent here. What number theory problem could be represented there? Here is the image after $n = 10^5$ iterations. So we have $\big\{ (x, T^n (x)) : x = \frac{m}{10^5} \text{ and }0 < m < 10^5 \big\}$. This looks like it might be mixing. Here is after a million ($n = 10^6$) iteratios. There's no formula for the quantative mixing of exchange maps.

Finite Galois group of Galois extension implies that the extension is finite? Posted: 20 Jun 2021 07:46 PM PDT Assume that the field extension $K \subset L$ is a Galois (in other words: normal and separable) extension with finite Galois group. Is it possible to prove, preferably without the use of Zorn's lemma, that the extension must be also finite? My idea so far is to show that the degree of every irreducible polynomial over $K$ is bounded by the order of $\mathrm{Gal}(L/K)$, let's say $n$, and hence the extension must be finite. The latter part is pretty easy to prove (under the condition of the former part), but how can the former part be proved? If I assume that an irreducible polynomial over $K$ has at least degree $n+1$, then it has at least $n+1$ distinct roots in $L$ (normality and separability). This means that there are at least $n+1$ possibilities to send such a root, let's say $a$, to another one. But how can such a monomorphism $K(a) \to L$ be extended to $L \to L$, preferably without the use of Zorn's lemma, in order to get at least $n+1$ elements in $\mathrm{Gal}(L/K)$ to get a contradiction?

A tough integral $\int_0^{\infty}\frac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $ Posted: 20 Jun 2021 07:41 PM PDT I was working on an integral which I found on Quora. I simplified it a lot and ended up with this intgeral $$\int_0^{\infty}\dfrac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $$ I tried converting this into exponential form and using geometric series which ended up in this. $$2\sum_{k=0}^\infty (-1)^k\int_0^\infty \dfrac{e^{-(2k+1)\pi x}}{4x^2+1}\, \mathrm dx $$ I didn't try to solve this using exponential integral, as I am not that much familiar with it. Using Wolfram|Alpha, I figured out that this integral is equal to $\frac{1}{2}\ln{2}$. The simple answer makes me suspect if the integral is just a tricky one. How can I evaluate this integral, using this method or any other method, except contour integration?

$\prod_{i=1}^{n} (3 +\frac{1}{a_i})$ cannot be a power of 2 if $a_i \equiv \pm 1 \pmod 6$ Posted: 20 Jun 2021 07:43 PM PDT I have encountered this problem and I can't solve it. If $a_i \equiv 1 \pmod 6$ or $a_i \equiv -1 \pmod \ 6$ and $a_i \ne \pm 1$, for every $i \in \{1, 2, .., n\}$, then prove that $$\prod_{i=1}^{n} \left(3 +\frac{1}{a_i}\right)$$ cannot be a power of $2$. Using induction, I proved that $$2^{n + 1} \lt \prod_{i=1}^{n} \left(3 +\frac{1}{a_i}\right)$$ Then I tried working$\mod 6$,$\mod4$ and $\mod 3$, but I couldn't solve it. Have you got any ideas?

How does folding paper affect the geometry angles? Posted: 20 Jun 2021 07:39 PM PDT This is how you create equilateral triangles. But I have noticed that if you start of with a flawed fold, you can still end up with equilateral triangles, how come? Instructions: Need: Rectangular strip paper. Step one make an arbitrary crease. Take the right side of the paper and align in to the crease, now fold, you should have another crease. Now alternate, fold up vertically then fold down. Code: Down, down, up, down, up, down... Why does any arbitrary crease lead to equilateral triangles.

Finding the orthogonal complement of a span? Posted: 20 Jun 2021 08:23 PM PDT Let $V = P_3(\mathbb{R})$ the vector space of all polynomials in $t$ of degree at most 3. $W = M_{2\times 2}(\mathbb{R})$ the vector space of all $2\times 2$ real matrices. Define $T:V \rightarrow W$ by $T(a_1 + a_2t + a_3t^2 + a_4t^3) = \begin{bmatrix} \dfrac{1}{\sqrt{1}}a_1 & \dfrac{1}{\sqrt{2}}a_2 \\ \dfrac{1}{\sqrt{4}}a_4 & \dfrac{1}{\sqrt{3}}a_3 \end{bmatrix}$. My question: Let $U = span\left(\left\{\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}, \begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}\right\}\right)$, find $U^\bot$. Is my attempt correct? Using Gram-schmidt process to get a set of orthonormal basis of $M_2$ with $u_1, u_2$ as given. Denoting $M$ as $a_{11}E_{11} + a_{12}E_{12} + a_{21}E_{21} + a_{22}E_{22}$, and denoting 2 matrices given as column vectors $(\alpha_1, \alpha_2)$, as the new matrtix $A$, solve $A'x = 0$ and orthonormalize the new basis, we get: \begin{align*} A = \begin{pmatrix} 1 & 1\\ 0 & 0\\ 0 & 2\\ 1 & 3 \end{pmatrix} \end{align*} Performing elementary row operations on $\begin{bmatrix} 1 & 0 & 0 & 1\\ 1 & 0 & 2 & 3 \end{bmatrix}$, we get the RREF: \begin{align*} \begin{bmatrix} 1 & 0 & 1 & 1\\0 & 0 & 1 & 0 \end{bmatrix} \end{align*} To find the null space, we solve the matrix equation \begin{align*} \begin{bmatrix} 1 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3\\x_4 \end{bmatrix} = \begin{bmatrix} 0\\0 \end{bmatrix} \end{align*} We take $x_2 = 2$, $x_4 = s$, then $x_1 = -s$, $x_3 = 0$. Hence, \begin{align*} \vec{x} = \begin{bmatrix} -s\\t\\0\\s \end{bmatrix} = \begin{bmatrix} 0\\1\\0\\0 \end{bmatrix} t + \begin{bmatrix} -1\\0\\0\\1 \end{bmatrix}s \end{align*} By solving $A^Tx = 0$, the orthogonal complement $U^\bot$ is: \begin{align*} \left\{\begin{bmatrix} 0\\1\\0\\0 \end{bmatrix},\begin{bmatrix} -1\\0\\0\\1 \end{bmatrix} \right\} \end{align*}

Derivative of $\mathrm{L}_2$ norm of function Posted: 20 Jun 2021 07:51 PM PDT I would like to find the derivative of the function $g$ with respect to $(\theta, z)$ where $y$ and $w$ are independent of $(\theta, z)$ and $\parallel\cdot\parallel_2$ denotes the $\mathrm{L}_2$ distance. $$ g(\theta, z) = \parallel f(\theta, z) - y\parallel_2-w $$ What is its gradient/jacobian? $$ \nabla_{\theta, z} \,\,g(\theta, z) = \,\,? $$

Does there always exist a field automorphism sending one root to another? Galois Theory Posted: 20 Jun 2021 08:21 PM PDT I don't really have an example in mind and this is just something I thought of. Say $L:K$ is a Galois extension and so $L:K$ is a splitting extension of some separable poly say $f\in K[X].$ Suppose $f$ has roots $\{a_1,\cdots\ a_n\}$ in $L$, I was wondering if I want to fix two roots, say $a_i,a_j$, does there always exist an field automorphism fixing $K$ thats sends $a_i$ to $a_j$? In other words, is it guaranteed that there exists an element in $Gal(L|K)$ such that it sends $a_i$ to $a_j$ for any choices of roots? Many thanks in advance! Edit: The answer is given in the link from Michael, in particular this works if and only if $f$ is irreducible.

Intuition behind this Olympiad problem using generating functions Posted: 20 Jun 2021 08:24 PM PDT From here https://web.archive.org/web/20160418085410/https://db.math.ust.hk/notes_download/elementary/algebra/ae_A11.pdf Example 4.1. (CGMO 2004) A set of poker has $32$ cards. $30$ of them are in red, yellow and blue, with $10$ cards in each colour, given the numbers $1, 2, …, 10$ respectively. $2$ of them are different jokers, both with the number $0$. A card with the number $k$ counts for $2^k$ points. Call a group of cards a 'good group' if the sum of their points is $2004$. Calculate the number of 'good groups' They define: $$G(x) = (1+x)^2(1+x^2)^3(1+x^4)^3...(1+x^{1024})^3$$ I cannot figure out how this was constructed exactly. What is the logic behind building this in this way?

Testing whether the cost-cutting measures seem to be working at the $5\%$ significance level Posted: 20 Jun 2021 08:06 PM PDT Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of $80$ dollars. New cost-cutting measures were started and a sample of $25$ claims was tested. The sample mean of the costs to process these claims was $76\%$ and the sample standard deviation of the costs was $\$10$. We would like to test whether the cost-cutting measures seem to be working at the $5\%$ significance level. State the null and alternative hypotheses for this test. Here is the $t$-distribution table $$\begin{array}{r|cccccccccccc} \text{cum. prob} & t_{.60} & t_{.76} & t_{.80} & t_{.85} & t_{.90} & t_{.95} & t_{.975} & t_{.99} & t_{.995} & t_{.999} & t_{.9995} \\ \text{one-tail} & 0.50 & 0.25 & 0.20 & 0.15 & 0.10 & 0.05 & 0.025 & 0.01 & 0.005 & 0.001 & 0.0005 \\ \text{two-tails} & 1.00 & 0.50 & 0.40 & 0.30 & 0.20 & 0.10 & 0.05 & 0.02 & 0.01 & 0.002 & 0.001\\ \hline \text{df} & & & & & & & & & & &\\ 1 & 0.000 & 1.000 & 1.376 & 1.963 & 3.078 & 6.314 & 12.71 & 31.82 & 63.66 & 318.31 & 636.62\\ 2 & 0.000 & 0.816 & 1.061 & 1.386 & 1.886 & 2.920 & 4.303 & 6.965 & 9.925 & 22.327 & 31.559\\ 3 & 0.000 & 0.765 & 0.978 & 1.250 & 1.638 & 2.353 & 3.182 & 4.541 & 5.841 & 10.215 & 12.924 \\ 4 & 0.000 & 0.741 & 0.941 & 1.190 & 1.533 & 2.132 & 2.776 & 3.747 & 4.604 & 7.173 & 8.610\\ 5 & 0.000 & 0.727 & 0.920 & 1.156 & 1.476 & 2.015 & 2.571 & 3.365 & 4.032 & 5.893 & 6.869\\\hline \color{blue}{6} & 0.000 & 0.718 & 0.906 & 1.134 & 1.440 & 1.943 & 2.447 & 3.143 & 3.707 & 5.208 & 5.959\\ \color{blue}{7} & 0.000 & 0.711 & 0.896 & 1.119 & 1.415 & 1.895 & 2.365 & 2.998 & 3.499 & 4.785 & 5.408\\ \color{blue}{8} & 0.000 & 0.706 & 0.889 & 1.108 & 1.397 & 1.860 & 2.306 & 2.896 & 3.355 & 4.501 & 5.041\\ \color{blue}{9} & 0.000 & 0.703 & 0.883 & 1.100 & 1.383 & 1.833 & 2.262 & 2.821 & 3.250 & 4.297 & 4.781\\ \color{blue}{10} & 0.000 & 0.700 & 0.879 & 1.093 & 1.372 & 1.812 & 2.228 & 2.764 & 3.169 & 4.144 & 4.587\\\hline 11 & 0.000 & 0.609 & 0.876 & 1.088 & 1.363 & 1.796 & 2.201 & 2.718 & 3.106 & 4.025 & 4.437\\ 12 & 0.000 & 0.695 & 0.873 & 1.083 & 1.356 & 1.782 & 2.179 & 2.681 & 3.055 & 3.930 & 4.318\\ 13 & 0.000 & 0.694 & 0.870 & 1.079 & 1.350 & 1.771 & 2.160 & 2.650 & 3.012 & 3.852 & 4.221\\ 14 & 0.000 & 0.692 & 0.868 & 1.076 & 1.345 & 1.761 & 2.145 & 2.624 & 2.977 & 3.787 & 4.140\\ 15 & 0.000 & 0.691 & 0.866 & 1.074 & 1.341 & 1.753 & 2.131 & 2.602 & 2.947 & 3.733 & 4.073\\\hline \color{blue}{16} & 0.000 & 0.690 & 0.865 & 1.071 & 1.337 & 1.746 & 2.120 & 2.583 & 2.921 & 3.686 & 4.015\\ \color{blue}{17} &0.000 & 0.689 & 0.863 & 1.069 & 1.333 & 1.740 & 2.110 & 2.567 & 2.898 & 3.646 & 3.965\\ \color{blue}{18} & 0.000 & 0.688 & 0.862 & 1.067 & 1.330 & 1.734 & 2.101 & 2.552 & 2.878 & 3.610 & 3.922\\ \color{blue}{19} &0.000 & 0.688 & 0.861 & 1.066 & 1.328 & 1.729 & 2.093 & 2.539 & 2.861 & 3.579 & 3.883\\ \color{blue}{20} & 0.000 & 0.687 & 0.860 & 1.064 & 1.325 & 1.725 & 2.086 & 2.528 & 2.845 & 3.552 & 3.850\\\hline 21 & 0.000 & 0.686 & 0.859 & 10.63 & 1.323 & 1.721 & 2.080 & 2.518 & 2.831 & 3.527 & 3.819\\ 22 & 0.000 & 0.686 & 0.858 & 1.061 & 1.321 & 1.717 & 2.074 & 2.508 & 2.819 & 3.505 & 3.792\\ 23 & 0.000 & 0.685 & 0.858 & 1.060 & 1.319 & 1.714 & 2.069 & 2.500 & 2.807 & 3.485 & 3.768\\ 24 & 0.000 & 0.685 & 0.857 & 10.59 & 1.318 & 1.711 & 2.604 & 2.492 & 2.797 & 3.467 & 3.745\\ 25 & 0.000 & 0.684 & 0.856 & 1.058 & 1.316 & 1.708 & 2.060 & 2.485 & 2.787 & 3.450 & 3.7325\\\hline \color{blue}{26} & 0.000 & 0.684 & 0.856 & 1.058 & 1.315 & 1.706 & 2.056 & 2.479 & 2.779 & 3.435 & 3.707\\ \color{blue}{27} & 0.000 & 0.684 & 0.855 & 1.057 & 1.314 & 1.703 & 2.052 & 2.473 & 2.771 & 3.421 & 3.690\\ \color{blue}{28} & 0.000 & 0.683 & 0.855 & 1.056 & 1.313 & 1.701 & 2.048 & 2.467 & 2.763 & 3.408 & 3.674\\ \color{blue}{29} & 0.000 & 0.683 & 0.853 & 1.055 & 1.311 & 1.699 & 2.045 & 2.462 & 2.756 & 3.396 & 3.659\\ \color{blue}{30} & 0.000 & 0.683 & 0.854 & 1.055 & 1.310 & 1.697 & 2.042 & 2.457 & 2.750 & 3.385 & 3.646\\\hline 40 & 0.000 & 0.681 & 0.851 & 1.050 & 1.303 & 1.684 & 2.021 & 2.423 & 2.704 & 3.307 & 3.551\\ 60 & 0.000 & 0.679 & 0.848 & 1.045 & 1.296 & 1.671 & 2.000 & 2.380 & 2.660 & 3.232 & 3.460\\ 80 & 0.000 & 0.677 & 0.845 & 1.042 & 1.290 & 1.660 & 1.984 & 2.364 & 2.626 & 3.174 & 3.390\\ 100 & 0.000 & 0.675 & 0.842 & 1.037 & 1.282 & 1.646 & 1.962 & 2.330 & 2.581 & 3.098 & 3.300\\ 1000 & 0.000 & 0.674 & 0.842 & 1.036 & 1.282 & 1.645 & 1.960 & 2.326 & 2.576 & 3.090 & 3.291\\\hline \color{blue}{\text{z}} & 0.000 & 0.674 & 0.842 & 1.036 & 1.282 & 1.645 & 1.960 & 2.326 & 2.576 & 3.090 & 3.291\\\hline \text{Confidence Level} & 0\% & 50\% & 60\% & 70\% & 80\% & 90\% & 95\% & 98\% & 99\% & 99.8\% & 99.9\%\\ \end{array}$$

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