Recent Questions - Mathematics Stack Exchange |
- Order list based on incomplete, pairwise, fuzzy comparisons
- 2-D heat equation with a source function general solution
- How can I interpret the 2D advection equation?
- Show that E[ln(X/E[X])] < $0$
- Euler characteristic of odd dimensional manifold - Hatcher
- Intuitively understanding why $\pi_1(S^2) = 0$
- Investigating improper integral convergence
- If $u$ is continuous on $\bar D$, harmonic on $D$ and vanishes on an open arc in $\partial D$, then is $u=0$?
- How to solve random variable example with absolute value function?
- Show that $\int_{0}^{\infty}\frac{|\sin(x)|}{|x|}dx=\infty$.
- What functions satisfy $\int_a^b f(x) c(x) \, dx \ge 0$ for all convex functions $f$?
- Prior in variational autoencoders
- Proof why solution of linear diophantine equation $ax-by=1$ is found by sign change of solution for $b$ in solution of $ax+by=1$
- Wording questions regarding the two pizzas 4 sizes and 8 toppings questions
- $\int_0^t (t-r)^{1/3} r^{5/3} dr$
- When is a system robust or volatile? Evaluating std deviations
- Number of ways of selecting $4$ people out of $12$ sitting on a round table such that no two of them are consecutive
- Can a complex Radon measure be approximated by compactly supported Radon measures?
- Why is $\phi_\mu(x)= \int_{\widehat{G}} \xi(x)d\mu(\xi)$ a continuous map.
- Moore-Penrose pseudoinverse solves the least squares problem (svd framework)
- Prove or disprove that $x_0\in\operatorname{cl} Y$ iff there exists a sequence $x_n$ in $Y\subseteq X$ converging to $x_0$ when $X$ is sequential.
- How to show that the set $A=[a,b)$ is not open nor closed?
- $f: $ Open set $U\subset \mathbb{R^m}\to\mathbb{R^n}$. $f$ is $C^1\iff$ directional derivatives continuous
- what about the case $V_x=U_x$ in the proof of paracompactness Theorem (John Lee book)?
- Why are single numbers sometimes written in brackets, as in $(2x)+(-4) +(x)$? [closed]
- How many squares of each colour are in a generalized checkerboard C-coloured m x n rectangle?
- (strong law of large numbers) We played a game in a casino. $X_i$ the money we won or lost the i-th time....
- Finding center of convex polygon
- A generalization of Jordan curve theorem to connected open sets in the plane
- Applications of the wreath product?
| Order list based on incomplete, pairwise, fuzzy comparisons Posted: 01 May 2022 08:13 AM PDT I have the following problem that I just cannot seem to find good references on.
The result is a list with pairwise comparisons that I might put in a matrix or something alike. Now my question is.. What methods/algorithms are there to do a best-effort ordering of the elements. It does not have to be perfect. |
| 2-D heat equation with a source function general solution Posted: 01 May 2022 08:10 AM PDT I am given a heat equation $F = \frac{\partial u}{\partial t}-\frac{1}{r}\frac{\partial }{\partial r} (r\frac{\partial u}{\partial r})-\frac{1}{r^2}\frac{\partial^2 u }{\partial θ^2} $ concurning $u(t,r,θ)$ and $F(t,r,θ)$ It has boundary and initial conditions $u(t, 1, θ) = 0$, $ u(0, r, θ) = 0$. and the equation is on a unit disk. The source function $F$ is continuous, bounded and obeys the same boundary conditions as $u$, and $u$ is bounded as $r$ -> 0. I need to write down the most general solution to this. All of my example questions don't have the heat equation in multiple spacial dimentions, or even in multiple dimensions period, so I'm not sure where to start. |
| How can I interpret the 2D advection equation? Posted: 01 May 2022 08:14 AM PDT I want to model the advection of debris rocks with a thickness h_d on top of a glacier through ice flow with velocity components u and v. Can anybody explain the difference between these 2 equations and which one I should take? Thanks
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| Posted: 01 May 2022 08:10 AM PDT We know that X > $0$ and that E[X] < $\infty$. |
| Euler characteristic of odd dimensional manifold - Hatcher Posted: 01 May 2022 08:03 AM PDT I ran into some trouble while reading through Hatcher's proof of the following:
There is only one part of the proof that I don't understand. He writes
I have been able to show that we indeed get $\mathbb{Z}_2$ summands in $H^i(M;\mathbb{Z}_2)$ (if helpful, I can edit this post to include this proof), but I fail to see that this is also the case in $H^{i-1}(M;\mathbb{Z}_2)$. My best guess is that it follows from the UCT, but I haven't been able to show it. Is there something obvious that I'm missing? |
| Intuitively understanding why $\pi_1(S^2) = 0$ Posted: 01 May 2022 07:57 AM PDT When we are saying that $\pi_1(S^2) = 0,$ are we speaking about a solid sphere or a hollow sphere? as far as I understand the fundamental group of a topological space is a measurement for the holes in $S^2,$ so by saying that $\pi_1(S^2) = 0,$ we mean that it has no holes? so we are speaking about a solid sphere? Could someone clarify this to me, please? |
| Investigating improper integral convergence Posted: 01 May 2022 07:56 AM PDT I'm trying to investigate the convergence of the following: $$ \int_{1}^{\infty} (1-cos(\dfrac{1}{x})) \,dx $$ Initially, its easy to see that the limit of $cos(\dfrac{1}{x})$ when $x\rightarrow \infty$ is $1$, therefor the the $(1-cos(\dfrac{1}{x}))$ goes to $0$ as $x\rightarrow \infty$, thus allowing me to deduce that the improper integral indeed converges. Yet I'm trying to prove this using the comparative/dirichlet/absolute convergence methods. I've tried playing around with trigonometric identities, substituting $1- cos(\dfrac{1}{x})$ with $2sin^2(\dfrac{1}{2x}))$ yet that didn't get me anywhere. Any assistance would be indeed helpful. On a different note- I'm having similar issues with $\int_{0}^{\infty}\dfrac{e^{2x}}{1+x^2}dx $, I've proved that $\int_{-\infty}^{1}\dfrac{e^{2x}}{1+x^2}dx $ converges but then our class lecturer decided that you cant use comparison tests with integrals in form of $\int_{-\infty}^{a}$ where $a\in \mathbb{R}$. Thanks in advance! |
| Posted: 01 May 2022 07:55 AM PDT Greene and Krantz, Chapter 7 ex. 11. Where $D$ is the unit disc. I have made some progess. I will list my results. If we assume that the answer is "no", I was able to deduce that any nonzero $u$ satisfying the other conditions must satisfy the following conditions also: $u$ is not holomorphic in $\bar D$, $u$ is real in $D$ and bounded in $\bar D$. In particular this implies that $u$ is real, nonconstant, and bounded in $D$. I have a feeling this somehow contradicts the max/min principal for harmonic functions, but I can't seem to see how exactly. I'm not entirely sure how to advance from here. There could be a counterexample I'm not seeing though. |
| How to solve random variable example with absolute value function? Posted: 01 May 2022 08:08 AM PDT pdf of x is $f_X(x)=$ $1\over2$exp$-(x+3)\over2$ (defind at x>-3) and Y=T(x) is y=|x| defined at (-1,1) and y= -1 for elsewhere. How can I get pdf of y $f_Y(y)=$ ? |
| Show that $\int_{0}^{\infty}\frac{|\sin(x)|}{|x|}dx=\infty$. Posted: 01 May 2022 08:10 AM PDT
How to solve this integral? I know the value of integral without the absolute value |
| What functions satisfy $\int_a^b f(x) c(x) \, dx \ge 0$ for all convex functions $f$? Posted: 01 May 2022 08:14 AM PDT This is an attempt to generalize Inspired by that question and the given answers, I have the following
The "only if" direction is easy: If $(1)$ holds for the four convex functions $x \mapsto \pm 1$, $x \mapsto \pm x$ then $(2)$ holds. So the interesting part is the "if" direction. In the above mentioned Q&A this has been proven for the functions $c(x) = \cos(k x)$ (on the interval $[0, 2 \pi]$). Some of the proofs given there use the fact that $$ \int_0^{2\pi} \cos (x) \, dx = \int_0^{2\pi} x \cos (x) \, dx = 0 \, , $$ but all proofs use also symmetries of the cosine function, trigonometric identities, or where the cosine is positive and negative in $[0, 2 \pi]$. My conjecture is that these additional properties of the cosine are not needed, and that $(2)$ alone is sufficient to prove $(1)$ for all convex functions $f$. There is a simple proof for the "if" direction under the additional assumption that $f$ is twice continuously differentiable: Let $c_1$ be an antiderivative of $c$, and $c_2$ be an antiderivative of $c_1$. By adding a constant to $c_2$, if necessary, we can assume that $c_2(x) \ge 0$ on $[a, b]$. If $(1)$ holds then $c_1(a) = c_1(b)$ and $c_2(a) = c_2(b)$, and for all convex functions $f$ on $[a, b]$ is, using integration by parts (twice): $$ \int_a^b f(x) c(x) \, dx = \int_a^b f''(x) c_2(x) \, dx \ge 0 \,. $$ What I am looking for is a proof of the conjecture (the "if" direction) which works for all convex functions $f$, without additional assumptions on differentiability. |
| Prior in variational autoencoders Posted: 01 May 2022 07:51 AM PDT I am currently dealing with variational autoencoders where I've read the original paper "An introduction to variational Bayes" from Kingma and Welling. I am currently still a little confused about the choice of the prior $p(z)$. For my understanding the variational autoencoder has the three possible distribution families we can choose:
Now for my understanding and also according to this notation used, the parameters for the decoder and the prior have to be the same, since overall we want to compute $p_\theta(x,z)$ with a neural network right? Then, how is it possible to choose $p_\theta(z)=\mathcal{N}(0,I)$ while at the same time optimising the parameters $\theta$ computed by the decoder network? As a third question: if not restricting the prior to be a standard normal distribution, how do we learn the parameters of the prior $p_\theta(z)$? |
| Posted: 01 May 2022 08:10 AM PDT Say, given LDE $113 x +42y=1$ have solution given by $$113 = 2.42 +29\implies 29= 113 - 2.42$$ $$42= 1.29+13 \implies 13=42 -1.29$$ $$29= 2.13 +3 \implies 3=29 -2.13$$ $$13= 4.3 +1\implies 1=13 -4.3$$ $$3= 3.1 +0$$ Writing in reverse. $$13 -4.3=1$$ $$13 -4.(29 -2.13)=1\implies -4.29 + 9.13= 1$$ $$-4.29 + 9.(42-29)=1\implies -13.29 + 9.42= 1$$ $$9.42 -13.(113-2.42)= 1\implies 35.42-13.113= 1$$ So, one solution is $(X,Y)= (-13, 35)$. Learned that for LDE $113 x -42y=1$, one solution $(X,Y)= (-13, -35)$ How it is obtained is unclear. Say, applying the same process to LDE $113 x -42y=1$ get: $$113 = (-2).(-42) +29\implies 29= 113 + 2.(-42)$$ $$(-42)= (-1).29+(-13) \implies -13=(-42) + 1.29$$ $$(-29)= 2.(-13) -3 \implies -3=(-29) -2.(-13)$$ $$(-13)= 4.(-3) -1\implies 1=(13)+4.(-3)$$ $$-3= -3.1 +0$$ Writing in reverse. $$13 +4.(-3)=1$$ $$13 +4.(-29 -2.(-13))=1\implies -4.29 + 9.13= 1$$ $$-4.29 + 9.(42-29)=1\implies -13.29 + 9.42= 1$$ $$9.42 -13.(113-2.42)= 1\implies 35.42-13.113=1$$ So, where erred in not getting $-35.42-13.113=1$ |
| Wording questions regarding the two pizzas 4 sizes and 8 toppings questions Posted: 01 May 2022 07:49 AM PDT Wording questions regarding the following question: "You are ordering two pizzas. A pizza can be small, medium, large or extra large, with any combination of 8 possible toppings (getting no toppings is also allowed, as is getting all of 8). How many possibilities are there for your two pizzas?" The above question and similar questions have being asked here in the past. And the numerical answers are as such: size choices = 4 size choices = $\binom{4}{1}$ multiplication rule: $\binom{4}{1}$ $\sum_{i=0}^8\binom{8}{i} = 4*256 = 1024$ First question: is it necessary to multiple 1024 by 2 even when both pizzas has the same combinations to address the original question? Second question, I've seen an answer where $2^8$ is used instead of $\sum_{i=0}^8\binom{8}{i}$, where the contributor wrote that "the number of possible combinations of toppings is the same as sampling from the set {0,1} with replacement and with ordering, 8 times; there are 2^8=256 possible toppings combinations". Which I don't understand where he derived "8 times". And am I right to assume "set {0,1}" means 0 = with, 1 = without topping and vice versa. Third question: There are also mentioning of using the Einstein-Bose approach for replacement where $\binom{1024+2-1}{2}$. Since the question has no mentioning of replacement, hence, should I be viewing this as a with or without replacement question? |
| $\int_0^t (t-r)^{1/3} r^{5/3} dr$ Posted: 01 May 2022 07:47 AM PDT The integral $$\int_0^t (t-r)^{1/3} r^{5/3} dr$$ came up in Miller's Introduction to Differential Equations (1987), (Sec 6.6, Convolutions, in the chapter on Laplace transformations, problem 5a on p. 328). Given the section, convoluted functions multiply under the Laplace transform, and etc. Okay, fine. But taken out of context as just another integral, I was stumped. If there were squared terms in the parentheses I could try trig substitutions. If the exponents were integers, fine. If it was just a single term with a fractional exponent, and not multiplied by that difference, okay. But this general form with fractional exponents, I'm not sure what to do. How do I solve this and other integrals like it? I'm going to try sticking a second question in here. Miller integrates his convolutions from 0 to t, other people seem to go from minus infinity to infinity. What's with that? (It's been a long time since I've used LaTeX, and I'm new to MathJax on the web, so I'm not sure how this is going to turn out.) |
| When is a system robust or volatile? Evaluating std deviations Posted: 01 May 2022 07:47 AM PDT Let's say I have a number of non-deterministic systems that produce numerical outputs as such: Input -> System 1 -> Output 1 Input -> System 2 -> Output 2 etc. Since the systems are non-deterministic, running multiple experiments for each one will produce different outputs. This allows us to calculate the standard deviation of the outputs of each system. A system with a low std deviation can be considered robust, while one with a high standard deviation of the outputs is considered volatile. My question is, where is that line drawn? Is a relative standard deviation of (e.g.) 12% considered low or high? Can I only compare the systems with regard to each other or is there a rule for measuring where the line is drawn? Is there another way to evaluate the systems with respect to the volatility/robustness of their outputs? Thank you very much for any insights on this |
| Posted: 01 May 2022 08:13 AM PDT Indexing the people as $x_i$ where $i\in(1,2,\cdots,12)$ So, each choice of $x_i$ must differ atleast by one and hence they are the solution to the equation. $x_1+1+x_2+1+x_3+1+x_4+1=12$ Which by the stars and bars method is $\binom{7}{3}$ which is apparently not correct, where am I going wrong here? |
| Can a complex Radon measure be approximated by compactly supported Radon measures? Posted: 01 May 2022 07:59 AM PDT Let $G$ be an (abelian) locally compact Hausdorff group. Consider the following fragment from Folland's text "A course in abstract harmonic analysis" (second edition, p102).
Why is the boxed line true? I guess formally, the statement we should prove is the following: If $\mu \in M(G)$, there exists a sequence $\{\mu_n\}_{n=1}^\infty \subseteq M(G)$ such that $\|\mu_n -\mu\|\to 0$ and such that $\mu_n$ has support on a compact set (i.e. there exists a compact subset $K_n\subseteq G$ such that $\mu_n(A)=\mu_n(A\cap K_n)$ for all Borel subsets $A\subseteq G$. Is this statement true? If so, how can we prove this? I have no idea how to construct such a sequence of measures (especially if $G$ is not assumed to be $\sigma$-compact, but even that simpler case remains elusive to me). |
| Why is $\phi_\mu(x)= \int_{\widehat{G}} \xi(x)d\mu(\xi)$ a continuous map. Posted: 01 May 2022 08:00 AM PDT Let $G$ be a locally compact Hausdorff abelian group. Let $\widehat{G}$ be its dual group, consisting of the unitary characters $G \to \mathbb{T}$. If $\mu \in M(\widehat{G})$ (= complex Radon measures on $\widehat{G}$), define $$\phi_\mu: G \to \mathbb{C}: x\mapsto\int_{\widehat{G}} \xi(x) d\mu(\xi).$$ Note that this integral exists because $$\int_{\widehat{G}}|\xi(x)|d|\mu|(\xi)= |\mu|(\widehat{G}) < \infty.$$ Why is the map $\phi_\mu$ continuous? I.e. assume that we have the net convergence $x_\alpha \to x$ in $G$. Then we should be able to show that $$\int_\widehat{G} \xi(x_\alpha) d \mu(\xi)\to \int_\widehat{G} \xi(x)d\mu(\xi).$$ I tried to estimate $$\left|\int_\widehat{G} \xi(x_\alpha)d\mu(\xi)-\int_\widehat{G} \xi(x)d\mu(\xi)\right|\le \int_\widehat{G}|\xi(x_\alpha)-\xi(x)| d|\mu|(\xi).$$ Now, I am not sure how to proceed. Any hints/help are highly appreciated! Reference: Folland's book "A course in abstract harmonic analysis", second edition p103. |
| Moore-Penrose pseudoinverse solves the least squares problem (svd framework) Posted: 01 May 2022 07:58 AM PDT So I'm a computer science researcher who has to learn some numerical linear algebra for my work. I've been struggling with the SVD and Moore-Penrose pseudoinverse as of late. I'm trying to solve some problems to get more comfortable with what should probably be routine manipulations. First of all, I have gone through similar questions on Stack Exchange but I believe they were more general and are not equivalent. I'm working in the framework where $A^{\dagger}=V\Lambda^{\dagger}U^T$. So basically I'm using SVD's. The matrix $A$ of course is identified with $U\Lambda V^T$. Here is the problem: Consider the matrix equation $Ax=y$, where $A\in R^{m\times n}$. The corresponding least squares problem is to find a least squares solution $x_{LS}$ that minimizes the Euclidean norm of the residual, i.e., $||Ax_{LS}-y||=min_{x\in R^n}||Ax-y||=min_{z\in Ran(A)}||z-y||$ a) Show that $A^{\dagger}y$ is a least-squares solution and satisfies the normal equation $A^TAx=A^Ty$. Why is this solution special? b) Show that $Ker(A^TA)=Ker(A)$. c) Use the above results to deduce that $x\in R^n$ is a least-squares solution if and only if it satisfies the normal equation. Help on any or all of these parts is appreciated. I'd also appreciate links to relevant posts. Like I said, I've read similar questions but did not understand them as they were in a more general framework. |
| Posted: 01 May 2022 07:51 AM PDT Hte following is an exercise from Elementos de Topología General by Ángel Tamariz Mascarúa and Fidel Casarrubias Segura. First a couple of definitions:
Problem 7 and 8 in that textbook say:
Here are the statement of the Propositions mentioned in the problem:
So I am really confused about the preceding theorem becasue if it is was true then it seem to me it would prove that a sequential space is a Fréchet-Uryshon space but as here showed this is false. So first of all to follow I put a direct reference of the text I mentioned hoping I understood bad Spanish text.
Anyway I tried to prove the statement and surprisingly it seems true. So to follow my proof attempt.
So is the proposition $3$.D.$8$.b.$1$ true? if it is true then is the proof I gave correct? Could someone help me, please? |
| How to show that the set $A=[a,b)$ is not open nor closed? Posted: 01 May 2022 08:09 AM PDT I only have the definition that a set $B$ is open if for all $x \in B$, there exists $\epsilon > 0 $ such that there is an open ball centered at $x$ and radius $\epsilon$ which is a subset of $B$. First, I want proof by contradiction to show that $A$ is not open. Suppose that it is open, thus I must let $x \in A$ and $\epsilon > 0$. How do I show that there exists an open ball centered at $x$ and radius $\epsilon$ which is a subset of $A$? My main issue is how to get a good choice for $\epsilon$? Similarly, to show that $A$ is not closed, I want to show that its complement $A^c$ is open, by a definition. Is it enough to show that $(-\infty, a)$ is open? |
| Posted: 01 May 2022 07:45 AM PDT Let $f=(f_1,f_2,\cdots,f_n)$ be a function from an open set $U$ in $\mathbb{R}^m$ to $\mathbb{R}^n$. Now $f$ is $C^1$ iff all partial derivatives $D_1,D_2,\cdots,D_m$ of all the components of $f$ are continuous. Can we say the similar where instead of taking partial derivatives of the components of $f$ we take directional derivatives $D_{v_1},D_{v_2},\cdots,D_{v_k}$ of components of $f$. What should be the value of $k$ and do we need any other conditions ? Now my intuition is that for directional derivatives $k=m$ and $v_1,\cdots,v_m$ should be linearly independent. But i can not prove it. Can you help me |
| what about the case $V_x=U_x$ in the proof of paracompactness Theorem (John Lee book)? Posted: 01 May 2022 08:01 AM PDT Here the proof of "paracompactness Theorem" (Theorem 4.77) of the book "Introduction to topological manifold" - John M. Lee (2ed):
When $\mathcal{U} = \{W_j\}$ then $V_x = U_x \cap W_j = W_j = U_x$, and if the only subcover of $A_j$ from $W_j$ is $W_j$ itself, then the new constructed cover is just $U$, and we don't have refinement of $U$. Something wrong with my argument here? EDIT: please, check this anwser. Since $W_j$ is open, for each point of $W_j$ there's an open ball in $W_j$ contains that point, then $W_j$ is not only subcover of $A_j$. |
| Why are single numbers sometimes written in brackets, as in $(2x)+(-4) +(x)$? [closed] Posted: 01 May 2022 07:52 AM PDT Why, in some expressions, are numbers written in brackets? like $$(2x)+(-4) +(x)$$ |
| How many squares of each colour are in a generalized checkerboard C-coloured m x n rectangle? Posted: 01 May 2022 08:11 AM PDT Assume an $m\times n$ rectangle has been been divided into a grid of $mn$ unit squares, and the squares have been coloured with $C$ colours in such a way that the colours in any row or column cycle, i.e. if the colours are represented by the number $1,2,\dots , C$ then $1 \rightarrow 2$, $2 \rightarrow 3$, $\dots$ ,$C-1 \rightarrow C$, $C \rightarrow 1$, etc. With this setup there are actually $C$ possible coloured variants, so we choose the variant that has a black square in the $(1,1)$ position. An example for $m = 10$, $n = 13$ and $C = 4$ is illustrated below with its `matrix' representation: Here we have number of squares of colour $1 = 33$ number of squares of colour $2 = 33$ number of squares of colour $3 = 32$ number of squares of colour $4 = 32$ So what is the general formula? I'm hoping that this is a known problem and someone can point me in the right direction (even if it is an open problem). Thank you. |
| Posted: 01 May 2022 07:44 AM PDT
Not sure if i understand.But what did I think. $X_n = \begin{cases} +1, & p \\[2ex] -1, & q \end{cases}$ If $p = q = p - q = 0 \to \lim_{n\to\infty} \sum_{i=1}^n\frac{X_n}{n} = \lim_{n\to\infty} \sum_{i=1}^n\frac{0}{n}= 0 $ Now for the cases $p \gt q$ and $p \lt q$. Now for the cases $p \gt q$ and $p \lt q$, I understand that the sum is positive ($p \gt q$) and negative ($p \lt q$), but does it go to plus infinity and minus infinity? |
| Finding center of convex polygon Posted: 01 May 2022 08:01 AM PDT
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| A generalization of Jordan curve theorem to connected open sets in the plane Posted: 01 May 2022 08:03 AM PDT Problem(Fulton's Algebraic Topology: A First Course, Problem 5.23) Let $U\subseteq\mathbb R^2$ be any connected open set in the plane.
ThoughtsThe proof of Jordan's curve theorem on Fulton's book is based on Mayer-Vietoris sequence for de Rham cohomology group, which relies heavily on the fact that the universal space is $\mathbb R^2$ (or homeomorphic to it). If $U$ is not simply connected, it seems that I cannot simply apply the original proof. My idea is to find a neighborhood of $X$ which is homeomorphic to $\mathbb R^2$. I have no clear idea how to do that. Any idea? Thanks! |
| Applications of the wreath product? Posted: 01 May 2022 08:04 AM PDT We recently went through the wreath product in my group theory class, but the definition still seems a bit unmotivated to me. The two reasons I can see for it are 1) it allows us to construct new groups, and 2) we can use it to reconstruct imprimitive group actions. Are there any applications of the wreath product outside of pure group theory? |
If I'm given vertices of a convex polygon (in the attached image, they are D,E,F,G and H) if we know that inside the polygon there exists a point (say O) for which each angle created by any two adjacent two vertices and the O are equal. That means, angle DOE, angle DOH, angle HOG, angle GOF and angle FOE all are equal. How to find O? | 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 |
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