Suppose that you have a (finite) list of Boolean values and that you would like to count the number of occurrences of True. This (infinite) state machine's state-transition function is $$f (x, u) := \... View answer 1 votes$$ \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 ...

Given $\mathrm A \in \mathbb{R}^{n \times n}$ with no zero rows, we would like to find $\mathrm x \in \mathbb{R}^n$ such that every entry of $\rm Ax$ is nonzero. Let $\mathrm b \in \mathbb{R}^n$ be a ...

We have three binary variables $x, y, z \in \{0,1\}$ and the following if-then-else (ITE) condition $$\text{if } x = 1 \text{ then } y = 1 \text{ else } y = z$$ If $x = 1$, then $y = 1$ but $z \in \{... View answer 2 votes Let$\rm V$be the$2 \times n$matrix whose$i$-th column is vector$\rm v_i$. We have the Boolean optimization problem in$\mathrm z \in \{0,1\}^n$$$\min_{\mathrm z \in \{0,1\}^n} \| \mathrm V \... View answer 4 votes Simplifying the problem: Given a positive integer r positive integers m, n \geq 2 a partial binary matrix \mathrm A \in \{*, 0,1\}^{m \times n} (where * denotes an unknown entry)... View answer 1 votes To complement the 2nd part of D.W.'s answer, we would like to find an \mathcal H-polytope (defined by the intersection of closed half-spaces) whose intersection with \{0,1\}^4 is$$\{ (1,1,0,0),(... View answer Accepted answer 3 votes If$m \times m$symmetric matrix$\rm H^{\top} H$is circulant, then its spectral decomposition is $$\rm H^{\top} H = Q \Lambda Q^*$$ where the eigenvalues of$\rm H^{\top} H$are given by the ... View answer Accepted answer 2 votes Let$x \in [-10,10] \cap \mathbb Z$and$y \in \{0,1\}$. Suppose that $$y = \begin{cases} 1 & \text{if } x \geq 3\\ 0 & \text{if } x \leq 2\end{cases}$$ To ensure that$y \neq 0$when$x \...

Let $f : \{0,1,\dots,5\} \to \{0,1\}$ be defined by $$f (x) := \begin{cases} 0 & \text{if } x = 5\\ 1 & \text{if } x \neq 5\end{cases}$$ Let $y = f (x)$. We would like to maximize $y$. Note ...

Given $1 \leq m \leq n$, we want to determine the cardinality of the following set $$\{ \mathrm X \in \{0,1\}^{n \times n} \mid \mathrm X 1_n \leq m 1_n \,\land\, \mathrm 1_n^{\top} \mathrm X \leq m ... View answer 7 votes In \mathbb R, one can simply round down or round up to obtain an element of \mathbb Z. Only two choices! However, in \mathbb R^n, one has 2^n ways of rounding to obtain an element of the ... View answer 3 votes Suppose we have the following deck of cards$$\{\mbox{Ace}, 2, 3, 4, 5, 6, 7, 8\}$$I shuffle the deck, pick a card, but do not show it to you. You are totally ignorant as to what card I picked. ... View answer 3 votes$$a \lor (\neg a \land b) \equiv (\underbrace{a \lor \neg a}_{\equiv \text{True}}) \land (a \lor b) \equiv \text{True} \land (a \lor b) \equiv a \lor b