A farmer has a rectangular ground of 100 m by 50 m, he wants to plant olive trees, in sufficiently spaced ways (to avoid exhaustion by the roots) at least 10 meters from each other. How much can one hope to put at the most, effectively?

A map $f\colon\mathbb{Z}\longrightarrow\mathbb Z$ is called a quasi-homomorphism if the set$$\{f(m+n)-f(m)-f(n)\,|\,m,n\in\mathbb{Z}\}$$is bounded. Let $R$ be the set of these functions. Let's consider the binary relation $\sim$ in $R$ defined by$$f_1\sim f_2\iff\{f_1(n)-f_2(n)\,|\,n\in\mathbb{Z}...

i am currently studying Stochastic Processes without Measure Theory. I have a question on Gambler's Ruin problem. Suppose each round of game are independent bernoulli trials $X_k$ for $k\geq 0$, whereby the player wins $\$1$ with probability $p$ and lose $\$1$ with probability $1-p$. $P(X_{k+1}...

Let $U$ and $V$ be the null spaces of $ A=\begin{bmatrix} 1&1&0&0\\0& 0&1&1 \end{bmatrix} $ and $ B=\begin{bmatrix} 1&2&3&2\\0&1&2&1 \end{bmatrix} $. Then what will be the dimension of $U+V.$ I calculated the null space of $U$ and $V$ as follows: \begin{align*} U=\{x\in \mathbb{R}^4: Ax=0 \}\im...

Consider a gambling process $(X_n)_{n∈\mathbb{N}}$ on the state space $S = {0, 1, . . . , N}$, with probability $p$, resp. $q$, of moving up, resp. down, at each time step. For $x = 0, 1, . . . , N$, let $τ_x$ denote the first hitting time, $τ_x := \inf\{n ≥ 0 : X_n = x\}$ Let $p_x := P(τ_{x+1} <...