Recent Questions - Mathematics Stack Exchange

---

• Is there a function that satisfies the following conditions?
• How does an element of $\Bbb{Q}\otimes_\Bbb{Z}A$ looks like?
• determinant of the sum of a matrix and an indicator matrix
• Is $d(x,y)$ = $\sqrt{| x - y |}$ over $\mathbb{R}$ complete metric space?
• How do you calculate the possibilities of an alphanumeric string?
• $M$ Poisson r.v. with parameter $\lambda T(F(b)-F(a))$
• Clarification with Ito's Lemma and holomorphic functions
• No. of possible ways to answer 2018 questions
• Requesting suggestion for software to solve polynomial equation with coefficients in GF(1024)
• The vertex forms of polynomials, and why quadratics are not the end. what about quartics?
• If order of $H$ equals order of $x$ in $H$ then $x$ generates $H.$
• A nice countable family of open subsets of $\mathbb{R}$
• time complexity and big O notation of sub set dynamic programming
• Calculate $\int_0^{\infty} |v(y)| (\int_y^{\infty} e^{-(Im z )(x-y)} dx )dy$ for $Im (z) < 0$ and $\int_0^{\infty} |v(y)| dy = \|v\|$
• Solution to $\int^{\infty}_0 \frac{1}{q} e^{-aq^2} dq.$
• Diagonalization of a matrix given by a finite group
• Help proving a set equality - excercise from the book of proof third edition
• finding the smalest element of a function defined by an integral
• Prove that the line is parallel
• Show that if $u_n:=p(f(x_n),g(y_n))$ for all $n\in \mathbb{N}$, then the sequence $(u_n)$ converges.
• Moving denominator $x$ to numerator for algebra fraction
• Does the quadratic formula hold modulo $n$ for $ax^2 + bx + c \equiv 0 \pmod n$?
• Is every Zariski-closed real matrix Lie group the joint stabilizer of some list of mixed tensors?
• Which Dynkin diagram is being spoken about here? Why is there a double line?
• Finding the area of a triangle on a unit circle
• Using Pythagoras Theorem, find the value of $x$ in this triangle. [closed]
• Prove that $DG+HE=GH$ for the given figure.
• Prove that $\lim_{x\to\infty}\frac{1}{x}\int_{0}^xf(t)\,dt=a$ if $f$ is continuous and $\lim_{x\to\infty}f(x)=a$
• Is there any intuitive understanding of normal subgroup?

Is there a function that satisfies the following conditions? Posted: 08 Jun 2022 09:12 AM PDT I want to know if there is a function $R = f(n,m,k)$ -- where $n, m$ and $k$ are integer values that can only be positive or $0$ -- that satisfies the following conditions: If $n=0$, $k>0$; for any $m$, $R=1$. If $m=0$, $k>0$; for $n>0$, $R=0$. If $m>0, n>0, k>0$; then $0<R<1$. If $k=0$, then $R$ takes on all values between $0$ and $1$. In other words, if $k=0$, $R$ cannot be narrowed down to a specific number within $[0, 1]$. The last condition has me stumped, and I don't know how to express that as a mathematical structure or idea. It's pretty difficult to create a function that satisfies the other three as well. Any insight you might have is welcome.

How does an element of $\Bbb{Q}\otimes_\Bbb{Z}A$ looks like? Posted: 08 Jun 2022 09:13 AM PDT I have the functor $~~~\Bbb{Q}\otimes_\Bbb{Z}-~~~$ from $\Bbb{Z}$-modules to $\Bbb{Q}$-vectorspaces. If I cosider a left exact sequence of $\Bbb{Z}$-modules $$0\rightarrow A\stackrel{f}{\rightarrow} B\rightarrow C$$and then apply the functor I get $$0\rightarrow \Bbb{Q}\otimes_\Bbb{Z}A\stackrel{id_\Bbb{Q}\otimes f}{\rightarrow} \Bbb{Q}\otimes_\Bbb{Z}B\rightarrow \Bbb{Q}\otimes_\Bbb{Z}C$$ First I only want to check that $\ker(id_\Bbb{Q}\otimes f)=im(0\rightarrow \Bbb{Q}\otimes_\Bbb{Z}A)$. So more precisly I want to check that $\ker(id_\Bbb{Q}\otimes f)=0$. But my problem is that I don't see how an element looks like in $\Bbb{Q}\otimes_\Bbb{Z}A$. I mean I know that in $u\otimes v\in U\otimes V$ where $(u,v)\in U\times V$. But is it here the same so does an element looks like $q\otimes a$ for $(q,a)\in \Bbb{Q}\times A$? Because in the lecture our prof somehow wrote something like $\sum_i q_i\otimes a_i$. Why do we need a sum? Is this more useful to prove injectivity? And is my idea with $q\otimes a$ wrong or does it also works with this representation? It would be nice if someone could explain this to me a bit in detail since I'm really confused. Thanks for your help.

determinant of the sum of a matrix and an indicator matrix Posted: 08 Jun 2022 09:06 AM PDT I don't know how to prove the following result and I would like to know if you have some ideas: Let $M \succ 0 \in \mathbb{R}^{n \times n}$ with $M_P=[M_{i,j}]_{i,j \in P}$ for $P \subseteq \{1,...,n\}$. Then: \begin{align*} \sum_{Q \subseteq P \subseteq \{1,...,n\}} \det(M_P)=\det(M+I_{\overline{{Q}}}) \end{align*} With $I_{\overline{Q}} \in \mathbb{R}^{n \times n}$ the diagonal matrix with $[1_{\overline{Q}}(i)]_{1 \leq i \leq n}$ as diagonal (indicator matrix of $\overline{Q}$). This is actually a generalization of the following known result: \begin{align*} \sum_{P \subseteq \{1,...,n\}} \det(M_P)=det(M+I_n) \end{align*}

Is $d(x,y)$ = $\sqrt{| x - y |}$ over $\mathbb{R}$ complete metric space? Posted: 08 Jun 2022 09:05 AM PDT To show that $\mathbb{R}$ is a complete metric space over the given metric above I have to show that every Cauchy sequence is converge sequence I know that $\mathbb{R}$ over $d_1(x,y)$ $=$ | x - y | is a complete metric space . Any help please

How do you calculate the possibilities of an alphanumeric string? Posted: 08 Jun 2022 09:05 AM PDT I want to know how many combinations are possible with a 4 character 4 digit string. All capital. Ex. ABCD1234, AAAA0000 - ZZZZ9999. What's the answer but more importantly, what's the formula?

$M$ Poisson r.v. with parameter $\lambda T(F(b)-F(a))$ Posted: 08 Jun 2022 09:04 AM PDT I have problems with the following statement: Let $\tau_1,\tau_2,\ldots$ i.i.d. r.v. with distribution $\exp(\lambda)$, and $N_{t}$ the Poisson associated process. Let $X_1,X_2, \ldots$ continuos r.v. i.i.d. with distribution function $F : \mathbb{R} \to [0,1]$; also independent of the r.v. $\tau_k$'s. Difine the random variable $$M=\#\left\{n:\left(\sum_{k=1}^n \tau_k,X_n\right) \in [0,T] \times [a,b]\right\}.$$ Prove that $M$ is a Poisson random variable with parameter $\lambda T(F(b)-F(a))$. I need to prove that $$P[M=m]=\frac{(\lambda T(F(b)-F(a)))^m}{m!} e^{-\lambda T(F(b)-F(a))},$$ but I cannot come with that solution. A hint was given in the exercise. Hint: Calcule $P(M = m)$ using the total probability law. With this I think I need to make the following: $$\sum_{i=m}^{\infty} P[M=m,N_T=i],$$ but the problem came when I have to manipulate the random variable $M$. I cannot came with anything. I guess I have to write down an equivalent way of the random variable $M$. Also I think I have to use in some step that $\sum_{k=1}^n \tau_k \sim \Gamma(n,\lambda)$. Any help? Thanks in advance.

Clarification with Ito's Lemma and holomorphic functions Posted: 08 Jun 2022 09:00 AM PDT I have been reading this page about Ito's Lemma, and I thought about something. This question is more of a reference request that anything. In the easiest case, where we have a Brownian motion $B_t$ and consider a function real valued function $f(x,t)$ such that is $C^2$, then we know also how compute the differential of the new porcess $f(B_t,t)$. Suppose now that $f(x,t)$ is a nice holomorphic function on $x$ and $C^2$ on $t$. Does the usual form of the differential hold? Thanks in advance!

No. of possible ways to answer 2018 questions Posted: 08 Jun 2022 09:07 AM PDT In the planet 'Hoyto' every question has three answers. Yes, No, or Maybe; in the planet 'Sokto' every question has two answers. Yes or No. If you ask 2018 questions, what is the difference between the number of all possible ways of answering the questions between planet 'Sokto' and planet 'Hayto'? My answer: $3^{2018} - 2^{2018}$ It's a problem from BdMO 2013. I have no idea about how to approach this kind of problems ('cause I'm a beginner). Please check the ans .

Requesting suggestion for software to solve polynomial equation with coefficients in GF(1024) Posted: 08 Jun 2022 08:54 AM PDT For example, the equation $x^2 + x + 1 = 0$ has the solutions $\alpha^{682}$ and $\alpha^{341}$, where $\alpha$ is a primitive element of GF(1024). I am looking for any software that can solve these equations where the polynomial also has coefficients from GF(1024). I only need to get the roots of cubic or quadratic polynomials for now.

The vertex forms of polynomials, and why quadratics are not the end. what about quartics? Posted: 08 Jun 2022 09:05 AM PDT What even is a vertex form? A vertex form of a function is the general form of the function defined as the transformations of a parent function. 2 stretches and 2 dilations make 4 parameters to work with. A linear polynomial has 2 parameters in general form, quadratics have 3, cubics have 4, and then you're out of parameters. So before even trying, we know quartics and beyond can't work. Linear functions are free, literally nothing needs to be changed. $$cx+d=cx+d$$ With quadratics, we've been taught completing the square a million times over in school. In general $$bx^{2}+cx+d=b\left(x+\frac{c}{2b}\right)^{2}+\frac{4bd-c^{2}}{4b}$$ However, have you ever tried to complete the cube? Let's look back at our parameter logic. We now need all 4 parameters, and the parent function is the simplest function of its kind, in this case, $x^3$. But when we try, we run in to a problem. $$p\left(\frac{x-h}{q}\right)^{3}+k=\frac{p}{q^{3}}\left(x-h\right)^{3}+k$$ the stretches are redundant with each other. This is because $x^3$, or any other exponent, will distribute across a multiplication or division. $$\left(\frac{x}{q}\right)^{n}=\frac{x^{n}}{q^{n}}$$ With that said, doesn't that mean a cubic vertex form is impossible? Well, not quite. We assumed that $x^3$ is the simplest cubic, and is therefore the parent function. What cubic could possibly be simpler than $x^3$? We have to think outside the box. Remember these trigonometric rules? $$\cos\left(2x\right)=2\cos\left(x\right)^{2}-1$$ $$\cos\left(x+y\right)=\cos\left(x\right)\cos\left(y\right)-\sin\left(x\right)\sin\left(y\right)$$ Applying this recursively gives the Chebyshev polynomial. $$T_{n}=\cos\left(n\cos^{-1}\left(x\right)\right)$$. In the case of a cubic, we get an alternative parent function. $$\cos\left(3\cos^{-1}\left(x\right)\right)=4x^{3}-3x$$ This time, the horizontal and vertical stretches are not redundant! $$a\cos\left(3\cos^{-1}\left(\frac{x}{b}\right)\right)\ne c\cos\left(3\cos^{-1}\left(x\right)\right)$$ Finally... $$ax^{3}+bx^{2}+cx+d=\frac{2\sqrt{\left(b^{2}-3ac\right)^{3}}}{27a^{2}}\cos\left(3\cos^{-1}\left(\frac{x+\frac{b}{3a}}{\frac{2\sqrt{b^{2}-3ac}}{3a}}\right)\right)+\frac{2b^{3}-9abc+27a^{2}d}{27a^{2}}$$ Except... there is 1 more extension to be made. While quartics can't be expressed in vertex form, they might be expressable with just 1 x. In particular, $$sx^{4}+ax^{3}+bx^{2}+cx+d\ ≟\ \frac{p}{f\left(\frac{x-h}{q}\right)+j}+k$$

If order of $H$ equals order of $x$ in $H$ then $x$ generates $H.$ Posted: 08 Jun 2022 09:11 AM PDT If order of $H$ equals order of $x$ in $H$ then $x$ generates $H.$ My attempt was: assume $y \in H$ such that $y \neq x^{k}$ $\forall 1 \leq k \leq n$ we know that order of y should be some integer $\leq n $ say t then $y^{t} = x^{n}$ and also we know that $xy \in H$ then order of xy $\leq n $ say m then $(xy)^m = e$ $\implies y^{m} = x^{-m}$ now using this how can I arrive at a contradiction? Is there some other way of proving the same?

A nice countable family of open subsets of $\mathbb{R}$ Posted: 08 Jun 2022 09:08 AM PDT My question is the following (under $\mathtt{ZFC}$), where I assume that the topology of $\mathbb{R}$ is the usual one: Is there a countable family $(U_n)_{n\in\mathbb{N}}$ of non-empty open subsets of $\mathbb{R}$ such that for every $X\subseteq \mathbb{N}$ which is infinite and co-infinite, there is a an $x\in\mathbb{R}$ with $\{n\in\mathbb{N}\mid x \in U_n\} = X$? Why? Why not? Ideas? Thanks!

time complexity and big O notation of sub set dynamic programming Posted: 08 Jun 2022 08:54 AM PDT given set of $n$ positive integers and a target number $T$ there is an dynamic programming algorithm that run in $O(n T)$ time complexity that solves the sub-set sum problem, It is regarded as exponential in terms of $T$, when $T$ is a big number, for the sake of this question assume that there is an algorithm that solves sub-set sum in $O(n \ln {T})$ time complexity, does the algorithm still runs in exponential time and why ? or does it runs in polynomial time(that will put sub-set sum in P) and why ? I am leaning toward the polynomial time algorithm since you need $O(\ln {T})$ time at least for writing $T$ digits in the memory !! I might be very wrong !! Thanks in Advance.

Calculate $\int_0^{\infty} |v(y)| (\int_y^{\infty} e^{-(Im z )(x-y)} dx )dy$ for $Im (z) < 0$ and $\int_0^{\infty} |v(y)| dy = \|v\|$ Posted: 08 Jun 2022 09:12 AM PDT This is the crucial part where I am stuck, because it should be $\int_y^{\infty} e^{-(Im z )(x-y)} dx = C$ so it'll be $C \cdot \|v\|$. That's the whole exercise, we need to show that $R(z;D)$ is a bounded operator for $Im(z) < 0$ in $L_1$ space: $$\|{R(z; D)v(x)}\| = \int_0^{\infty} |\int_x^{\infty} -i e^{iz(x-y)} v(y) dy|dx \le \int_0^{\infty} (\int_x^{\infty} |-i e^{iz(x-y)} v(y)| dy )dx =$$ $$= \int_0^{\infty} (\int_x^{\infty} |-i| |e^{iz(x-y)}| |v(y)| dy )dx = \int_0^{\infty} (\int_x^{\infty} e^{-(Im z )(x-y)}|v(y)| dy )dx =$$ $$= \int_0^{\infty} |v(y)| (\int_y^{\infty} e^{-(Im z )(x-y)} dx )dy$$ where $$R(z; D)v(x) = -i \int_x^{\infty} e^{i z(x-y)} v(y) dy, \quad x \in \mathbb{R}_+ \ a.e.$$

Solution to $\int^{\infty}_0 \frac{1}{q} e^{-aq^2} dq.$ Posted: 08 Jun 2022 08:57 AM PDT My apologies if this question has already been asked. I've tried using the search function but could not find the answer. I'm looking to solve the following integral: $$ \int^\infty_0 \frac{1}{q} e^{-aq^2} dq, $$ where $a >0 $ and $q$ denotes the magnitude of the momentum. Now obviously this integral does not converge on the given domain but I am looking for find a way to extract a sensible approximation from this equation. The context is that I am calculating decoherence rates for various quantum mechanical interactions and when calculating the decoherence rate for møller scattering I am left with a bunch of constants and this integral. I would really appreciate any thoughts on this. Thank you in advance!

Diagonalization of a matrix given by a finite group Posted: 08 Jun 2022 09:01 AM PDT Let $G$ be a finite group and $S : G \to \C$ be any map. Define the matrix $$ M = (S(xy^{-1}))_{x, y \in G} $$ Is it true that $M$ is diagonalizable of $\C$ ? Note: it would be equivalent to take $M' := (S(xy))_{x, y \in G} $. Thoughts: It is true if $G = \Z / g \Z$ is a cyclic group of order $g>1$, as one can check that for $U := (e^{2 \pi i x y / g})_{1 \leq x, y \leq g}$, we have $MU = UD$ for some diagonal matrix $D$. Moreover, $U$ is invertible (check the SAGE code below). This was in some paper by Takashi Ono. Apparently, it also works for some non-abelian groups, but I don't know how to proceed. UCF = UniversalCyclotomicField(); for g in range(1, 6): U = matrix(UCF, [[E(g)^(x*y) for x in range(g)] for y in range(g)]) print(g, U.determinant())#, CC(U.determinant())) print(U, "\n") print(U.inverse(), " ====== \n\n")

Help proving a set equality - excercise from the book of proof third edition Posted: 08 Jun 2022 09:14 AM PDT I am going through The Book of Proof - Third Edition, and came across this exercise: Do you think the statement $(\mathbb{R}-\mathbb{Z})\times\mathbb{N}=(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})$ is true, or false? Justify. I think the intent of the question was for me to answer graphically, since proving set equality was not introduced yet, but I attempted to do it anyway. Can someone offer a constructive feedback on my technique? Thanks in advance. $\mathbf{Ans.}$ The statement is true. Proof: Let $k\in (\mathbb{R}-\mathbb{Z})\times\mathbb{N}$. Then, $k=(x_1,n_1):x_1\in\mathbb{R},x_1\notin\mathbb{Z},n_1\in\mathbb{N}$. This means $k\in(\mathbb{R}\times\mathbb{N})$ and $k\notin(\mathbb{Z}\times\mathbb{N})$. Thus $(\mathbb{R}-\mathbb{Z})\times\mathbb{N}\subseteq(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})$. Now, let $l\in(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})$. Then, $l\in\mathbb{R}\times\mathbb{N}$ and $l\notin\mathbb{Z}\times\mathbb{N}$. This means $l$ is of the form $(x_2,n_2):x_2\in\mathbb{R},x_2\notin\mathbb{Z},n_2\in\mathbb{N}$. Thefore, $l\in(\mathbb{R}-\mathbb{Z})\times\mathbb{N}$. Thus, $(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})\subseteq(\mathbb{R}-\mathbb{Z})\times\mathbb{N}$. We conclude $(\mathbb{R}-\mathbb{Z})\times\mathbb{N}=(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})$. It feels to me i might have omitted some steps along the way, but I am not sure how to improve the proof further.

finding the smalest element of a function defined by an integral Posted: 08 Jun 2022 09:09 AM PDT what is the smallest element of the set? $$\{\int_{0}^1 (x^2 - ax - b)^2 dx: (a,b) \in \mathbb{R}^2\}?$$ See this image I tried setting the derivative equal $0$ but it is not working.

Prove that the line is parallel Posted: 08 Jun 2022 08:57 AM PDT Let $ABC$ be a non-equilateral triangle and $\omega$ be the inscribed circle of triangle $ABC$, which touches sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$, respectively. Let $G$ be the point on the circle $\omega$ such that $\angle AGD=90^\circ$. If $DG$ and $EF$ intersect at point $P$, prove that $AP$ is parallel to $BC$. I'm trying to prove that $\angle CAP=\angle ACB$. By diameter of incircle lemma, let $AG$ meet $BC$ at point $X$, then $X$ is the point where the excircle of triangle $ABC$ touches $BC$. But I really have no idea how to start. Let AG intersects ω at point H, then DH is the diameter of ω.

Show that if $u_n:=p(f(x_n),g(y_n))$ for all $n\in \mathbb{N}$, then the sequence $(u_n)$ converges. Posted: 08 Jun 2022 09:05 AM PDT Let $(X,d)$ and $(Y,p)$ be metric spaces, where $X$ is complete, and define continuous functions $f,g:X \to Y$. If $(x_n)$ and $(y_n)$ are Cauchy sequences in $X$ and $u_n:=p(f(x_n),g(y_n))$ for all $n\in \mathbb{N}$, show that the sequence $(u_n)$ converges. What I knew was: Since $f$ is continuous, say at $x_0\in X$, then for any $\varepsilon >0$, there exists $\delta _1>0$ such that for all $x\in X$ with $d(x,x_0)<\delta _1$, we have $p(f(x),f(x_0))<\varepsilon$. Similarly, since $g$ is continuous, say at $y_0\in X$, then for any $\varepsilon >0$, there exists $\delta _2>0$ such that for all $y\in X$ with $d(y,y_0)<\delta _2$, we have $p(f(y),f(y_0))<\varepsilon$. Since $X$ is complete, then $(x_n)$ and $(y_n)$ are convergent, say $x_n\to x$ and $y_n\to y$, where $x,y\in X$. This means that for any $\varepsilon >0$, there exists $N_1,N_2\in \mathbb{N}$ such that for all $n\geq N_1$ and $n\geq N_2$, we have $d(x_n,x)<\varepsilon$ and $d(y_n,y)<\varepsilon$, respectively. I didn't know yet how to apply these facts to showing that $(u_n)$ is converges. Any helps? Thanks in advanced.

Moving denominator $x$ to numerator for algebra fraction Posted: 08 Jun 2022 09:02 AM PDT For the question, $x=\frac{a}{{3b}}$, can the denominator of "$b$" be brought up to the numerator such that it becomes $x=\frac{a/b}{3}$ ? If not, what are the laws that I have to apply? The question I am solving is $x=\frac{a}{b} = \frac{b}{a/3}$.

Does the quadratic formula hold modulo $n$ for $ax^2 + bx + c \equiv 0 \pmod n$? Posted: 08 Jun 2022 08:54 AM PDT Does the quadratic formula $\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ hold modulo $n$ for $ax^2 + bx + c \equiv 0 \pmod n$? Computing the square root would require factoring $n$ and using either special-case formulas or the Tonelli-Shanks algorithm, but does the Quadratic Formula hold if used in this way? (For composite $n$, more than two square roots are possible, and that'd need to be accounted for.)

Is every Zariski-closed real matrix Lie group the joint stabilizer of some list of mixed tensors? Posted: 08 Jun 2022 09:00 AM PDT Consider a finite-dimensional real vector space $V$, and an embedded real Lie subgroup $G \subset \mathrm{GL}_\mathbb{R}(V)$. In what follows, $V^*$ denotes the real dual vector space of $V$. Def: Let us say $G$ has property $\mathcal{P}$ if there exist some finitely many $\alpha_1,\ldots,\alpha_k$ in the mixed real tensor algebra $TV$ (including tensor factors of both $V$ and $V^*$) of $V$, such that $G= \bigcap_{j=1}^{k}\mathrm{Stab}_{\mathrm{GL}_{\mathbb{R}}(V)}(\alpha_j)$. Here we use the induced action of $\mathrm{GL}(V)$ on the mixed tensor algebra $TV$. My question is whether every Zariski-closed Lie subgroup of $\mathrm{GL}_{\mathbb{R}}(V)$ has property $\mathcal{P}$? (Here we use the Zariski topology on $\mathrm{End}_{\mathbb{R}}(V)$ coming from real polynomials, e.g. from a choice of basis, or just in the multilinear sense.) Examples having property $\mathcal{P}$: For $V = \mathbb{R}^n$, for $G = \mathrm{O}(n,\mathbb{R})$, we can use $\alpha \in V^* \otimes V^*$ given by the standard real-bilinear inner product. For $V = \mathbb{R}^n$, for $G = \mathrm{SL}(n,\mathbb{R})$, we can use $\alpha \in V^{\otimes n}$ the standard volume form. For $V = \mathbb{R}^{2n}$, for $G = \mathrm{GL}(n,\mathbb{C}) \subset \mathrm{GL}(2n,\mathbb{R})$, we can use $\alpha = J = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right) \in V \otimes_\mathbb{R} V^*$. For $V = \mathbb{R}^{2n}$, for $G = U(n)$, if I'm not mistaken, we can use both $\alpha = J = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right) \in V \otimes_\mathbb{R} V^*$, and the $\beta_1,\beta_2 \in V^* \otimes_\mathbb{R} V^* $ which correspond to the real and imaginary parts of the standard Hermitian inner product $\mathbb{C}^{n} \times \mathbb{C}^{n} \rightarrow \mathbb{C}$ but treated as a real-bilinear map. For $V = \mathbb{R}^n$ and $G = \mathbb{R}^\times 1 \subset \mathrm{GL}(n,\mathbb{R})$ the nonzero multiples of the identity, we should be able to take any $\alpha_1,\ldots,\alpha_{n^2} \in V \otimes_\mathbb{R} V^* \simeq \mathrm{M}(n,\mathbb{R})$ such that the $\alpha$'s form an $\mathbb{R}$-linear basis of $\mathrm{M}(n,\mathbb{R})$. (This uses the fact that only the scalar multiples of the identity commute with all other square matrices.) Lemma: If $G$ has property $\mathcal{P}$, then $G \subset \mathrm{GL}_{\mathbb{R}}(V)$ is Zariski-closed; i.e. it is the intersection of $\mathrm{GL}_{\mathbb{R}}(V)$ with a Zariski-closed subset (a real polynomial-vanishing subset) of $\mathrm{End}_{\mathbb{R}}(V)$. Proof: After picking bases, each "stabilizing equation" $g \cdot \alpha_j = \alpha_j$ can be converted into a polynomial equation in the entries of $g$ and $g^{-1}$, and the $g^{-1}$ factors can be multiplied out to get a polynomial in just $g$'s entries; thus $G$ is the vanishing set in $\mathrm{GL}_{\mathbb{R}}(V)$ of a collection of real polynomials of $\mathrm{End}_{\mathbb{R}}(V)$. Rmk: For instance, as pointed out in the comments below, $\mathrm{GL}_+(n,\mathbb{R})$ is not Zariski-closed in $\mathrm{GL}_(n,\mathbb{R})$, hence $\mathrm{GL}_+(n,\mathbb{R})$ cannot have property $\mathcal{P}$. My question is: does every Zariski-closed, real Lie subgroup of $\mathrm{GL}_{\mathbb{R}}(V)$ have property $\mathcal{P}$? (I've updated the question to reflect the suggestions noticing that every $G$ having property $\mathcal{P}$ is Zariski-closed; now I'm wondering about the converse.)

Which Dynkin diagram is being spoken about here? Why is there a double line? Posted: 08 Jun 2022 08:59 AM PDT I'm confused about the following comment in Knapp's Lie Groups 2ed, page 397. Here, $\Delta$ is a root system associated to a complex semisimple Lie algebra, $\alpha, \beta$ are orthogonal roots and we also have that $\alpha \pm \beta \in \Delta$ (that is, they are not strongly orthogonal). I believe that the phrase simple component here means irreducible component. Since $\alpha \pm \beta$ are roots, $\alpha$ and $\beta$ must lie in the same irreducible component of $\Delta$. Further, since $\Delta$ comes from a semisimple Lie algebra, this irreducible component is a reduced root system (and hence the comment about only two root lengths). Question: What set of simple roots within (an irreducible component of) $\Delta$ is the author using to obtain the Cartan matrix/Dynkin diagram with double line? Attempt: I'm guessing there might be a way to show the existence of a simple system $\Pi \subset \Delta$ containing both $\alpha$ and $\beta - \alpha$. This would show that \begin{equation} 2\frac{(\alpha, \beta -\alpha)}{|\alpha|^2} \cdot 2 \frac{(\alpha, \beta-\alpha)}{|\beta-\alpha|^2} = 2, \end{equation} thereby giving the double line. But I don't see how to construct this $\Pi$. For example, if someone can provide a proof/reference showing that one can choose a simple system containing both $\alpha$ and $\beta - \alpha$, I will be happy.

Finding the area of a triangle on a unit circle Posted: 08 Jun 2022 09:11 AM PDT Assume that $0< \theta < \pi$. For three points $A(1,0)$, $B(\cos(\theta),\sin(\theta))$ and $C(\cos(2\theta),\sin(2\theta))$ on a unit circle, the area of triangle $ABC$ is: ??? I drew out the unit circle and tried to get the dimensions. I ended up drawing a triangle underneath the main triangle to try and get the base, but that ended up giving me a square root and I couldn't find a way to get rid of it. The answer is: $\sin(1-\cos)$

Using Pythagoras Theorem, find the value of $x$ in this triangle. [closed] Posted: 08 Jun 2022 09:06 AM PDT Using Pythagoras Theorem [Baudhāyana Śulbasûtra], find the value of $x$ in this triangle. I tried to solve it but I seem to get stuck with end result of $$26x+5x^2=588$$

Prove that $DG+HE=GH$ for the given figure. Posted: 08 Jun 2022 09:05 AM PDT What I could gather: $$\measuredangle ODB=\measuredangle OFC=\measuredangle OEC =90 \text{ degrees}$$ $$OD=OF=OE=r$$ $$\measuredangle ODE=\measuredangle OED$$ $$\measuredangle DOE=2\measuredangle DME$$

Prove that $\lim_{x\to\infty}\frac{1}{x}\int_{0}^xf(t)\,dt=a$ if $f$ is continuous and $\lim_{x\to\infty}f(x)=a$ Posted: 08 Jun 2022 09:08 AM PDT Suppose $f$ is continuous and $\displaystyle \lim_{x \to \infty} f(x) = a$. Prove that $$\displaystyle \lim_{x \to \infty}\dfrac{1}{x} \int_{0}^x f(t) dt = a.$$ I don't see why in the solution below they take the integral from $1$ to $N$ instead of $0$ to $N$ and how that proves the result. Also is $M$ considered to be positive? Book solution:

Is there any intuitive understanding of normal subgroup? Posted: 08 Jun 2022 08:59 AM PDT As the define goes: A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, for each element $n$ in $N$ and each $g$ in $G$, the element $gng^{−1}$ is still in $N$. My Question is: Can anyone give me an intuitive explaination or an example of this concept? Why it is very important in algebra?

You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now.	Email delivery powered by Google
Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States