We know that $\gcd(x, y) = d$ as d divides $x$ and $y$, now suppose there are $x’$ and $y’$ integers such that $$x = d \cdot x’ \implies d|x \\y = d \cdot y’ \implies d|y$$ then $a \cdot x$ would be $$ a\cdot x = a \cdot d \cdot x’$$ and so $a […]

This question is related to this question. Define $h(x)=|x|$ on $[-1,1]$ and extend it to $\mathbb R$ by defining $h(2+x) = h(x)$. This is a sawtooth function that is $0$ at even and $1$ at odd integers. Furthermore define $h_n(x) = (1/2)^n h(2^n x)$ and $$ g(x) = \sum_{n=0}^\infty {1\over 2^n }h(2^n x) = \sum_{n […]

A friend asked me the following question: “In an experiment, we are tossing a fair coin 200 times. We say that a coin flip was a success if it’s heads. What is the chance for having at least 6 consecutive successes?” And according to him, the answer is nearly 100%. My calculations were different. I’d […]

Let $T$ the transpose map $T(A)=A^t$ for $A\in M(2,\mathbb{R})$. I want to find a basis such that $T$ is diagonal. I considered $T$ as a map from $R^4\rightarrow R^4$ where $T$ can be represented by $$ \begin{pmatrix} 1 & 0 & 0& 0\\0&0&0&1\\0&0&1&0\\0&1&0&0\end{pmatrix} $$ The characteristic polynomial is $(t-1)^3(t+1)$ and by finding basis for each […]

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 – \{(0,0)\})$ but then realised that for better understanding I had to look at different examples, too. For the purpose of this question I am only interested in finding just one […]

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of a conformal map, which is conformal. Firstly, let’s get a conformal map from the disk to the unit disk, […]

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ is the one below. I used an algorithm (don’t know if you know this, but $\beta_0=\sqrt{141}+\lfloor\sqrt{141}\rfloor, \quad\beta_{n+1}=\frac{1}{\beta_n-\lfloor\beta_n\rfloor}$ $a_n=\lfloor\beta_n\rfloor$ $p_n=p_{n-1}a_n+p_{n-2}, \quad q_n=q_{n-1}a_n+q_{n-2} $) to determine the continued fraction expansion of […]

This question is giving me a lot of trouble. It’s taking a very long time to solve. My Work I’ll find how many strings have consecutive zeros and then deduct that from the total amount of strings Case I: The Length of String is $1$ $0$ of these strings have consecutive $0$ Case II: The […]

I want to see if my solution to the following problem in Ahlfors’ Complex Analysis text is correct. The problem reads: Let $\Omega$ be a doubly connected region whose complement consists of the components $E_1, E_2$. Prove that every analytic function $f(z)$ in $\Omega$ can be written in the form $f_1(z)+f_2(z)$ where $f_1(z)$ is analytic […]

Let $A \subset X$ and $B \subset Y$. Show that in the space $X \times Y$, $$\overline{A \times B}= \overline{A} \times \overline{B}$$ $\subset$: let $x \in \overline{A \times B}$. Then for every open set $W$ containing $x$, $W \cap (A\times B) \neq \emptyset$ Since $W$ is an open set of $X \times Y$, $W= U […]

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